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1
A General Valuation Framework
for Variable Annuities
To be used in the Option Interpolation
Model of Ortec Finance
Willeke de Tree
Master’s Thesis to obtain the degree in Econometrics (track: Financial Econometrics) University of Amsterdam
Amsterdam School of Economics Faculty of Economics and Business
Author: Ms. C.W. de Tree MSc Student nr: 10215832
Email: willekedetree@gmail.com
Date: July 31, 2017
University supervisor: Mr. prof. dr. H.P. Boswijk Second reader: Mr. dr. S.A. Broda
Company supervisors: Mr. ir. S.N. Singor &
Statement of Originality
This document is written by Willeke de Tree MSc who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
The aim of this thesis is to develop a general framework in which variable annuities with their specific characteristics can be valued from the point of view of an insurance company, and to analyse the sensitivity of the present value of a variable annuity con-tract with respect to multiple concon-tract specifications and the main risk drivers, namely insurance risk, market risk and behavioural risk.
In the first part of this thesis the theoretical research is described which provides an overview of variable annuities, their characteristics and the risk drivers involved. Furthermore, an investigation of the different methods to value variable annuities is carried out here. In the second part, a general framework is developed using risk-neutral Monte Carlo scenarios and a two-factor Hull-White Black-Scholes model for the financial market. Moreover, sensitivity analyses are performed to explore the sensitivity of the present value of a variable annuity contract with respect to the contract specifications and the risk drivers at time zero and over time.
The results of the sensitivity analyses show that not all contract specifications and risk drivers have the same influence on the present value of the variable annuity contract. According to the valuation at time zero six contract specifications and risk drivers are the most influential: 1) the interest rates, 2) the stock price volatility, 3) the lapse rates, 4) the growth factor of the benefits base, 5) the policyholder’s age, and 6) the asset mix. Increasing the interest rates, the percentage invested in stocks or the policyholder’s age has a positive effect on the average liability provision, while increasing the stock price volatility or the lapse rates or using a roll-up benefits base has a negative effect. Furthermore, the valuation over time adds the policyholder’s gender, and indirectly the mortality rates, to the list. Changing the policyholder’s gender has a positive effect on the average liability provision.
Keywords GMAB, GMDB, GMIB, GMWB, guarantees, Option Interpolation Model, valua-tion, variable annuities
Contents
Preface vii
1 Introduction 1
2 Variable annuities 3
2.1 Variable annuities . . . 3
2.1.1 Guarantees in variable annuities . . . 4
2.1.2 Growth of the benefits base . . . 5
2.2 Variable annuity fees . . . 6
2.3 Risk drivers . . . 7
2.3.1 Insurance risk . . . 7
2.3.2 Market risk . . . 8
2.3.3 Behavioural risk . . . 9
3 Valuation methods 10 3.1 Time zero valuation . . . 10
3.1.1 Single option rider valuation methods . . . 10
3.1.2 Unifying valuation methods . . . 12
3.2 Valuation over time . . . 13
3.2.1 Nested Monte Carlo simulation . . . 13
3.2.2 Least-squares Monte Carlo simulation method . . . 14
3.2.3 Option Interpolation Model . . . 15
4 Methodology 16 4.1 Specifications in the contract . . . 16
4.1.1 Decisions of the policyholder . . . 16
4.1.2 Decisions of the insurance company . . . 17
4.2 Policyholder behaviour . . . 18
4.3 The financial market . . . 18
4.4 Modelling the contract . . . 21
4.5 Evolution of the contract . . . 23
4.6 Present value of the contract . . . 27
5 Results 28 5.1 Results using fixed input parameters . . . 28
5.1.1 The example contract . . . 28
5.1.2 Comparison of growth factors and policyholder behaviour . . . . 29
5.1.3 Comparison of policyholder’s age and gender . . . 32
5.1.4 Comparison of asset mixes . . . 33
5.2.1 Guaranteed Minimum Death Benefit . . . 35
5.2.2 Guaranteed Minimum Accumulation Benefit . . . 35
5.2.3 Guaranteed Minimum Income Benefit . . . 37
5.2.4 Guaranteed Minimum Withdrawal Benefit . . . 37
5.3 Sensitivity analysis of the main risk drivers . . . 39
5.3.1 Sensitivity analysis of the interest rates . . . 39
5.3.2 Sensitivity analysis of the stock price volatility . . . 40
5.3.3 Sensitivity analysis of the mortality rates . . . 41
5.3.4 Sensitivity analysis of the lapse rates . . . 42
5.4 The contract over time . . . 43
6 Conclusion 46
Bibliography 49
Appendix A Overview of parameters and best-estimate cash flow vectors 52
Appendix B Overview of economic scenarios 54
Appendix C Further results using fixed input parameters 56 Appendix D Further results of the sensitivity analysis 62
Preface
In front of you lies the Master’s Thesis “A General Valuation Framework for Variable Annuities - To be used in the Option Interpolation Model of Ortec Finance”. It has been written to fulfil the graduation requirements of the Master Econometrics in the track Financial Econometrics. From April to July 2017 I have been engaged in the research for and the writing of this thesis.
In this thesis I have developed a general framework to value variable annuities with different characteristics from the point of view of an insurance company. Furthermore, I have created an example contract and examined the sensitivity of the contract with respect to the characteristics of the contract itself but also with respect to the main risk drivers. As I previously successfully have finished the Master Actuarial Science and Mathematical Finance at the University of Amsterdam, I wanted to combine my actuarial knowledge with my newly learned skills in Econometrics. In consultation with Ortec Finance I have found the subject of variable annuities, as Ortec Finance wished to examine the possibility of valuing variable annuities in its Option Interpolation Model and I was interested in developing and researching a general valuation framework.
I would like to thank the people without whom I could not have realised this thesis. First, I would like to thank my supervisors at Ortec Finance, Stefan Singor and Gerda Kalkhoven, for helping me to find a suitable subject, for guiding me through the pro-cess and for the critical questions during this propro-cess. Second, I would like to thank my supervisor at the University of Amsterdam, Peter Boswijk, for the advice and critical notes during my research and for the corrections during the writing of this thesis. Last, I would like to thank my parents, my sister, my partner and my friends for their support during the past months.
I hope you enjoy reading this thesis. Willeke de Tree MSc,
Chapter 1
Introduction
During the last decades variable annuity markets have developed in the United States and Japan providing retirement planning through diverse types of financial guarantees and living benefits (The Geneva Association, 2013). Already introduced for the first time in the United States by the Teachers Insurance and Annuities Association-College Retirement Equity Fund (TIAA-CREF) in 1952, the variable annuity market only gen-uinely developed in the early 1990s. Starting from variable annuity sales of around$25 billion in 1993 to a peak of more than $180 billion sales in 2006, the variable annuity market has become a substantial component of the life insurance market in the United States. In 2016 the variable annuity sales in the United States totalled over$100 billion, almost equal to half of the total annuity sales (Insured Retirement Institute,2017). The Japanese variable annuities market started evolving in the early 2000s and reached over U4,000 billion sales already in 2005 (The Geneva Association, 2013). After an exten-sive decline during the financial crisis, that started in 2008, the variable annuity sales have increased back to U1,390 billion in 2015 (The Life Insurance Association of Japan,
2016). Due partly to the increasing popularity in the United States and Japan, this type of annuity also has become up and coming in Europe in recent years, creating both opportunities and difficulties for the European insurance market.
To calculate the price of a variable annuity at present poses no major problems and general valuation methods are available to do so, see for example the article of
Bauer, Kling & Ruß (2008). In addition, a lot of studies have further developed and extended these valuation methods. However, under reporting and controlling guidelines such as the Own Risk and Solvency Assessment (ORSA) of Solvency II, the valuation of variable annuities must also be performed over time, which makes the process much more difficult due to the various guarantees involved (Milliman,2013). Valuing variable annuities over time in its basic form is performed using a nested Monte Carlo simula-tion method: firstly, a Monte Carlo simulasimula-tion is executed to obtain possible real-world scenarios in which the annuity must be priced and, secondly, for each point in time of each real-world scenario a risk-neutral Monte Carlo simulation is executed to obtain the value of the variable annuity at that particular time point (Cathcart & Morrison,2009). The problem is that this method is very time consuming and computer intensive ( Mil-liman,2013). Therefore, proxy methods have been proposed to value variable annuities over time, such as the least-squares Monte Carlo (LSMC) method which has originally been proposed byLongstaff & Schwartz(2001) to value American stock market options. Ortec Finance has developed the Option Interpolation Model to estimate the value of financial products (Ortec Finance,nd; Singor, Schols & Oosterlee,2016). The method
uses a Monte Carlo simulation to generate real-world scenarios. Then, instead of per-forming a nested Monte Carlo simulation, an interpolation method is used to determine the value of the financial product in each point in time of the real-world scenarios. The Option Interpolation Model interpolates on a grid of values of the financial product that has been calculated beforehand using a risk-neutral Monte Carlo simulation, where the values in the grid should represent all possible values of the financial product at each point in time in each real-world simulation. For multiple financial products this method already has proved to be far less time consuming than the nested Monte Carlo simu-lation method while remaining highly efficient. Since variable annuities become more popular in Europe, the described method can also be used to value variable annuities. However, before this method can be applied, a complete and comprehensive framework to value variable annuities must be developed. This framework must not only be ap-plicable for all types of variable annuities, but it also has to take into account the risk drivers of these products in a proper way. Furthermore, this framework should not only be able to calculate the value of variable annuities at present but should also be able to calculate the distribution of the value at future times.
This research aims to develop this unique framework to value variable annuities within the Option Interpolation Model, taking proper account of the different char-acteristics of the variable annuities and their risk drivers. For this purpose multiple sub-questions will be answered in this thesis. Firstly, the theoretical research will pro-vide an overview of variable annuities, their different characteristics and the main risk drivers affecting them alongside with an investigation of the different methods of valu-ing variable annuities. Secondly, in the empirical research a framework to value variable annuities at time zero will be developed keeping in mind the different characteristics and risk drivers. In order to value variable annuities, this research uses a Monte Carlo simulation to create risk-neutral scenarios to value the variable annuity properly. In the Monte Carlo simulation different economic scenarios will be generated using a two-factor Hull-White Black-Scholes model. The valuation of variable annuities will be based on the articles ofBauer et al.(2008) andBacinello, Millossovich, Olivieri & Pitacco(2011). Next to the valuation at time zero, the framework will be extended to value the variable annuity over time. Lastly, an analysis will be performed to explore the effect on the value of the variable annuity when changing the various contract characteristics or the main risk drivers.
The sequel of this thesis is constructed as follows. Chapter 2 reviews the theoretical background of variable annuities. It presents an overview of the different types of guar-antees, the fees associated with variable annuities and the risk drivers affecting variable annuities. Next, in Chapter 3 different methods for the valuation of variable annuities found in the literature are described. This chapter considers both methods for time zero valuation as well as methods for valuation over time and also clarifies the Option Inter-polation Model. Chapter4provides the methodology, including the valuation framework and necessary data, used in this thesis. The next chapter, Chapter 5, describes the val-uation results and the effect of the risk drivers on these valval-uation results found in the empirical research. Lastly, Chapter 6presents the conclusion of this thesis.
Chapter 2
Variable annuities
This chapter serves as a theoretical background for variable annuities. Firstly, a short overview of variable annuities is presented in Section2.1, including the guarantees the variable annuities are known for. Next, Section 2.2 describes the fees associated with variable annuities which the policyholder has to pay to the insurance company. Lastly, Section2.3 deals with the possible main risk drivers affecting variable annuities.
2.1
Variable annuities
An annuity is a contract between an insurance company and a policyholder in which the insurance company agrees to pay an income stream to the policyholder for a specified period of time or for the lifetime of the policyholder. Different types of annuities exist which can be divided in two main groups: fixed annuities and variable annuities. Fixed annuities have predefined fixed benefit payments and therefore provide a stable future income. Variable annuities are byEIOPA (2011)1 defined as “unit-linked life insurance
contracts with investment guarantees which, in exchange for single or regular premiums, allow the policyholder to benefit from the upside of the unit but be partially or totally protected when the unit loses value”. So variable annuities exist of a wide range of products which all have in common that their benefit payments are protected against risks, for example investment risk or mortality risk, by selecting one or more options out of a set of possible guarantees (Bacinello et al.,2011). Both groups of annuities are often used as post-retirement income or for other long-term income purposes.
Another distinction can be made regarding the starting time of the benefit payments. Annuities can be sold as immediate annuities or deferred annuities (The Geneva Associ-ation,2013). Immediate annuities start making benefit payments out of the investment account right away, meaning from the moment the contract is concluded. They therefore do not have a deferral or accumulation period and start with the withdrawal period. Deferred annuities on the other hand postpone the benefit payments to some later date, for example to the time of retirement. During the accumulation period the policyholder pays premiums which are added to the investment account of the annuity administered by the insurance company, while during the withdrawal period the policyholder receives benefit payments from this account. Due to pulling out cash amounts, possible invest-ment losses and payinvest-ment of the fees specified in the contract, the investinvest-ment account value reduces during the withdrawal period for both immediate and deferred annuities. If the total investment account value has been reduced to zero and the contract has not finished yet, the insurance company needs to make benefit payments to the policyholder
from its own capital. The period in which the insurance company has to finance the variable annuity using its own funds is called the insured period.
A variable annuity has two distinct product features which are present in each annuity: 1) the investment account, and 2) the guarantees (The Geneva Association,2013). The investment account is a basket of investment funds, also called sub-accounts, to which the policyholder allocates his premiums. During the life of the annuity the policyholder can, under certain constraints, change the allocation of the sub-accounts and contribute additional premiums. The insurance company keeps these investment accounts separate from their own capital in order to protect them against claims against the insurance company itself in the event of insolvency. The guarantees are calculated in reference to the benefits base, which is defined as the notional amount used to determine the amount of benefit payments to the policyholder. The guarantees are further elaborated on in Subsection 2.1.1, while the growth of the benefits base is detailed in Subsection2.1.2. Variable annuities have been introduced in the United States by the Teachers Insurance and Annuities Association-College Retirement Equity Fund (TIAA-CREF) in 1952 (The Geneva Association, 2013). Since then the market for variable annuities in the United States has developed and during the 1990s the modern variable annuities have evolved. According to analyst reports summarized in Ledlie, Corry, Finkelstein, Ritchie, Su & Wilson (2008) the main reasons for the popularity of variable annuities in the United States are equity exposure, longevity protection, transparency, flexibility, profitability and capital efficiency. The Japanese market for variable annuities started developing in the early 2000s with outstanding growing numbers of sales in the first few years. This enormous growth can be explained by a number of factors, such as demographic trends, the economic environment, the concern for risk, the savings culture and the deregulation in Japan in the 2000s (Ledlie et al.,2008). In the mid-2000s the variable annuities also started spreading through Europe (The Geneva Association,2013).
In variable annuities features of normal unit-linked life insurance contracts are, ac-cording to Bacinello et al. (2011), matched to the guarantees, which are features of participating life insurance contracts. Therefore, the guarantees are explicit and specific fees can be applied to meet their costs, which is in contrast to most participating poli-cies. The different types of fees are detailed in Section2.2. Furthermore,Bacinello et al.
(2011) state that variable annuities have the attraction of having dynamic investment opportunities, providing protection against financial risks for the policyholder, and the payment of the (remaining) benefits in case of early death.Ledlie et al.(2008) put for-ward that variable annuities can give a more efficient trade-off between risk and return than other types of annuities, such as unit-linked annuities, or than a combination of a conventional annuity and an income drawdown.
2.1.1 Guarantees in variable annuities
Various types of guarantees, called option riders, have come into existence during the past decades due to the different purposes of the annuities. These option riders are attractive as they provide different types of financial protection to the policyholder’s investment account by contract design (Kling, Ruez & Ruß, 2014). When valuating a variable annuity, the option rider can be seen as an exotic or path dependent option based on the notional amount determined in the contract.
Ben-efits (GMDBs), and 2) Guaranteed Minimum Living BenBen-efits (GMLBs) (Bauer et al.,
2008). The class of Guaranteed Minimum Living Benefits can be divided into three subclasses: 1) Guaranteed Minimum Accumulation Benefits (GMABs), 2) Guaranteed Minimum Income Benefits (GMIBs), and 3) Guaranteed Minimum Withdrawal Benefits (GMWBs). The Guaranteed Minimum Withdrawal Benefit generally has a predefined maximum number of withdrawals. An extension of this option rider is the Guaranteed Lifetime Withdrawal Benefit (GLWB), which has an unlimited number of withdrawals (Kling et al.,2014). Below one can find a short description of each type of option rider (Bauer et al.,2008;Ledlie et al.,2008;The Geneva Association,2013):
GMDB Guaranteed Minimum Death Benefits guarantee the return of the prin-cipal invested, paid as a lump sum, upon the death of the policyholder regardless of the performance of the investment account.
GMAB Guaranteed Minimum Accumulation Benefits guarantee the return of the principal invested, paid as a lump sum, on a specific maturity or anniversary date provided that the policyholder is alive but regardless of the performance of the investment account.
GMIB Guaranteed Minimum Income Benefits guarantee to take out a minimum income stream in the form of a life annuity from a specified point in time based on the greater of the investment account value and the benefits base. It should be noted that the option to annuitize the contract is a one-time irreversible decision.
GMWB Guaranteed Minimum Withdrawal Benefits guarantee to take out a min-imum income stream through regular withdrawals, which are specified as a percentage of the maximum of the investment account value and the benefits base, for a specified number of years.
GLWB Guaranteed Lifetime Withdrawal Benefits guarantee to take out a min-imum income stream through regular withdrawals, which are specified as a percentage of the maximum of the investment account value and the benefits base, for the lifetime of the policyholder.
Another option rider described in the literature by for example Bacinello, Biffis & Mil-lossovich(2010) is the surrender guarantee. This option rider guarantees the policyholder to receive at least a minimum surrender benefit in case of surrendering the contract and can be imposed by the regulator and consumer protection laws. The option rider is not considered in the empirical research of this thesis, as the policyholder always receives a fixed percentage of the investment account value in case of surrender of the contract. 2.1.2 Growth of the benefits base
As described before, the option riders are calculated in reference to the benefits base. The benefits base is defined as the notional amount which is used to determine the amount of benefit payments from the option rider to the policyholder (The Geneva Association, 2013). Without a growth factor, the benefits base is equal to the return of premiums, which is the amount of premiums paid net of any withdrawals. Including growth there are three different growth factors possible: 1) the roll-up, 2) the ratchet, and 3) the reset. The different growth factors are explained below (Bauer et al., 2008;
EIOPA,2011;The Geneva Association,2013):
an-num at an in advance specified (simple or compounded) interest rate. ratchet The ratchet, also called high watermark, compares the existing benefits
base with the investment account value at certain previous points in time and sets the benefits base equal to the investment account value if the investment account value at one of these time points is higher than the benefits base.
reset The reset is a comparison of the original investment account value and the current investment account value, where the benefits base is set equal to the highest of the two values. This growth factor is not implemented automatically, but is only used on the suggestion of the policyholder. The ratchet and the reset look much alike as they both adjust the benefits base to the investment account value at some previous point in time. However, the major difference between the two is that the benefits base can never decrease using a ratchet, while a decrease can happen using a reset. Furthermore, combinations of the growth factors of the benefits base can be offered to the policyholders. A commonly known example of such a combination is the maximum of a roll-up and a ratchet.
2.2
Variable annuity fees
Due to the fact that features of unit-linked life insurance contracts are matched to features of participating life insurance contracts in variable annuities, particular fees can be specified which the policyholder has to pay to the insurance company (Bacinello et al., 2011). The most commonly used fees described in the literature are the option rider fee, the administration fee and the surrender fee.
The option rider fee, also called guarantee fee, is exclusively intended for the costs associated with an option rider. This fee is required by the insurance company to be able to provide the option rider in the contract. According toKling et al.(2014) the option rider fee is determined as a fixed annual percentage of the investment account value and deducted from this account as long as the investment account value is positive. The fixed percentage can also be calculated based on the benefits base or on the single premium, depending on the specification in the contract.
The administration fee is a cost that should cover the ongoing administrative ex-penses the insurance company has to make to maintain the policy and the associated investment account (The Geneva Association, 2013). It is often calculated as a fixed percentage of the investment account value and also deducted from this account. This fee is often combined with a mortality fee and together they are called the mortality and expenses (M&E) fee.
The surrender fee is charged when a withdrawal of the investment account is or-dered by the policyholder which is not part of the GMWB (Bauer et al., 2008). Only the withdrawals up to a certain limit under the GMWB are free of the surrender fee, as these withdrawals are already charged with the option rider fee. The surrender fee is proportional to the amount withdrawn by the policyholder or to a fixed percentage of the investment account value. The fee is meant to compensate the insurance company for the costs already made for the policy (The Geneva Association,2013).
Next to above mentioned fees there may exist an annual investment management fee in the contract. This fee compensates the managers of the underlying sub-accounts for their services and is often calculated as a percentage of the investment account value
(The Geneva Association,2013). Also an initial commission or upfront acquisition cost can be charged by the insurance company for advisory and setting up the contract. The insurance company has different costs which have to be covered by this fee, such as the agent sales commissions, the marketing costs and the issue costs. As these fees are not directly related to variable annuities, they will not be considered in the empirical research of this thesis.
2.3
Risk drivers
The risks involved with variable annuities can be divided in three main categories ac-cording to The Geneva Association (2013): 1) insurance risk, 2) market risk, and 3) behavioural risk. Subsections 2.3.1,2.3.2 and 2.3.3elaborate further on these risk cat-egories. Next to the risks directly related to variable annuities, the insurance company also faces operational risk and reputational risk (Kling et al., 2014). However, these types of risk are not considered in the empirical research of this thesis as they do not relate only to variable annuities.
Next to above mentioned risks, also specifications in the contract can constitute a risk to an insurance company. The age and gender of the policyholder, together with the duration of the contract, will influence the length of time the insurance company is exposed to risks involved with making benefit payments and to the possibility of risks in the development of the investment account. Furthermore, the size of the contract, called the notional, the used option riders and the tax status of the policyholder can influence the development of the investment account and introduce additional risks. All risks are managed by the insurance company through a risk management process. This process consists of three phases: 1) risk identification, 2) risk assessment, and 3) the risk management actions (Bacinello et al.,2011). In the phase of risk identification the source of each risk is investigated. The risks are split up into several components, where among others a distinction is made between hedgeable and non-hedgeable components. Next, the risk assessment phase tries to identify interactions between the risks to be able to set up an integrated approach for modelling the risks. Finally, in the phase of the risk management actions specific risk management tools are chosen to control and fi-nance the risks. There are multiple tools to manage the risks, including product design, prudence in assumptions, risk pooling, natural hedges, a diverse balance sheet, asset liability management and reinsurance (The Geneva Association, 2013). Furthermore, stress scenarios can be executed to analyse the effect of single and combined shocks on the provision and risk management decisions of the insurance company.
2.3.1 Insurance risk
The most important risks in the category of insurance risk are longevity risk and mortal-ity risk. Longevmortal-ity risk is defined as the risk of having incorrect mortalmortal-ity assumptions which has as consequence that policyholders will live on average longer than expected (The Geneva Association, 2013). Mortality risk is the risk of not being able to know exactly how many of the policyholders contracted by the insurance company will survive up to certain time points.
Mortality risk can be split up into two different parts: 1) non-systematic or idiosyn-cratic mortality risk, and 2) systematic mortality risk (Fung, Ignatieva & Sherris,2014). If the number of policyholders contracted by the insurance company is high enough, an
insurance company can diversify away the idiosyncratic mortality risk according to
Bernard & Kwak (2016). However, systemic mortality risk, and equivalently longevity risk, cannot be eliminated by diversification since it arises due to the stochastic nature of the survival probabilities (Fung et al.,2014).
Although longevity risk cannot be diversified away, it can be controlled by an insur-ance company. According toThe Geneva Association (2013) an insurance company can limit the longevity risk in the product design by having age requirements or restrictions which limit the period of time benefit payments have to be paid. Furthermore, prudence in the model assumptions regarding mortality might prove itself useful. Also testing the mortality assumptions by a stress scenario analysis can be used as part of the product design and pricing process. Lastly, an insurance company can try to diminish longevity risk by a natural hedge with other lines of business or by reinsurance.
2.3.2 Market risk
Market risk, also called financial risk, is according to The Geneva Association (2013) the risk of obtaining adverse financial impact caused by changes in the (capital) market factors. An insurance company makes assumptions about the performance of the (cap-ital) market factors in the pricing process of variable annuities. However, the market factors can behave differently over time than assumed in the initial pricing process and this is the main source of market risk. In other words, market risk is the risk associated with not knowing the future development of financial assets. The most commonly known types of market risk are equity market risk, interest rate risk, credit risk and foreign exchange rate risk. The first two types are the primary risks associated with variable annuities and are considered in the empirical research of this thesis.
Equity market risk describes the risk associated with holding equity as an invest-ment and can be split up in two different risks: 1) equity return risk, and 2) equity volatility risk (Milliman,2013). A decline of the equity market will cause a decline in the value of the policyholder’s investment account, which will increase the exposure of the insurance company to the risk that the investment account value is insufficient to fund the level of guarantees concluded in the contract (The Geneva Association,2013). Furthermore, changes in the capital markets will cause changes in the reserves of the insurance company and in the capital the insurance company must hold to be able to meet its obligation to all its policyholders.
Interest rate risk is the risk arising from a change in the interest rate curve, as stated by Milliman (2013). An upward shift in the interest rate curve will cause an increased exposure of the insurance company to the risk that the investment account value is insufficient to fund the level of guarantees concluded in the contract. On the other hand, a decline in the interest rate curve will increase the present value of the long-term income guarantees the insurance company is obliged to pay to the policyholder.
If both types of risk are not managed or managed insufficiently, this will cause solvency risk or balance-sheet volatility for the insurance company (The Geneva Asso-ciation,2013). Therefore, it is important to take market risk into account properly. As market risk is not idiosyncratic, it cannot be diversified away by selling a high number of policies, but the risk can be managed through product design, asset liability manage-ment and natural hedges. For example, an insurance company can include investmanage-ment restrictions and investment fees in its contracts to limit the exposure to market risk or adjust the option rider features in the contracts. Also prudence in the pricing assump-tions and stress scenario analyses are useful tools to control market risk.
2.3.3 Behavioural risk
The Geneva Association(2013) describes behavioural risk, also called utilisation risk, as the risk that policyholders make decisions that are not in agreement with the assump-tions made by the insurance company. There are two main types of risks in the category behavioural risk: 1) persistency risk, and 2) benefit utilisation risk. Persistency risk, also called lapse risk or surrender risk, is the risk of a policyholder lapsing his contract. An insurance company will receive fees only as long as the policyholder decides to persist the contract. Also the length of time the insurance company has to provide the option rider is depending on the lapse rate of the policyholder. Benefit utilisation risk is the risk of a policyholder cancelling an option rider. This will also influence the period of time the insurance company will receive fees for the option rider and the period of time the insurance company has to provide the option rider.
Behavioural risk results from the fact that a policyholder has many choices accord-ing to Kling et al. (2014). For example, a policyholder has the choice of (partially) surrendering the contract, the decision whether or not and when to annuitize the vari-able annuity and the decision whether or not and how much to withdraw every year. Therefore, insurance companies assume sub-optimal policyholder behaviour when pric-ing the contracts. When assumpric-ing sub-optimal policyholder behaviour, the insurance company assumes that (at least some) policyholders do not behave in the way that would maximize the value of the insurance company’s liabilities arising from the option riders included in the contract.
Next to assuming sub-optimal policyholder behaviour, the insurance company has two manners to mitigate the behavioural risk of policyholders (The Geneva Association,
2013). First, the insurance company can mitigate the risk by product design. Incorpo-rating restrictions on for example the level of the benefits payments, the investment account and policyholder behaviour can reduce the size of behavioural risk. Further-more, as behavioural risk is idiosyncratic, the risk can also be made more predictable by risk pooling over a large group of policyholders.
Chapter 3
Valuation methods
After reviewing the background of variable annuities in the previous chapter, this chap-ter considers the valuation methods used to value variable annuities. Section 3.1 con-centrates on the valuation of variable annuities at time zero. The valuation methods for different types of guarantees found in the literature are discussed. Next, Section3.2 reviews methods for valuation over time. Also the Option Interpolation Model of Ortec Finance is explored in this section as a possible valuation method.
3.1
Time zero valuation
A lot of studies have explored valuing variable annuities at time zero and how to find the fair option rider fee. Most articles focus on one, or a limited number, type of option rider and fit a valuation method to this option rider. In Subsection3.1.1a short overview of the valuation methods found in the literature is given. As this thesis tries to find a general framework for valuing variable annuities for the Option Interpolation Model, the focus of this thesis is on the articles ofBauer et al. (2008) and Bacinello et al. (2011). Both articles develop a universal pricing method for the guaranteed benefits in variable annuities. Subsection3.1.2elaborates further on these pricing methods.
3.1.1 Single option rider valuation methods
As there is a lot of literature about valuing variable annuities for single option rid-ers, the overview below is not comprehensive. It only provides a selection of important or recent studies. Almost all studies perform valuation of variable annuities under a risk-neutral measure and most studies add additions or extensions to the risk-neutral valuation method.
Regarding Guaranteed Minimum Death Benefits (GMDBs),Milevsky & Posner (2001) and Bacinello (2003) use risk-neutral option pricing theory to investigate the value of the GMDB in variable annuities. They consider various death benefits, including re-turn of premium, rising floors and ratchets, and compute the fair fee charged to the investment account which is needed to fund the option rider. AlsoB´elanger, Forsyth & Labahn (2009) compute the fair fee associated with the GMDB, but allow for partial withdrawals. The results show that adding partial withdrawals significantly increases the fair fee. Furthermore, Ulm (2014) obtains a closed-form analytic solution for the GMDB using ratchets under a variety of mortality laws. Bernard, Hardy & MacKay
(2014) explore a new fee structure for variable annuities where the fee rate depends on the moneyness of the GMDB and investigate the effect of this fee structure on the
surrender decision. Next,Liang & Sheng(2016) study two types of GMDBs, namely the inflation guarantee and the combination guarantee, and construct a framework to price these options with stochastic inflation rates, interest rates and risky assets. Looking past risk-neutral valuation Marquardt, Platen & Jaschke (2008) propose a method for pricing GMDBs under a benchmark approach which does not require the existence of a risk-neutral probability measure. They in particular consider a minimal market model and find that the fair price of a roll-up GMDB is lower than the price obtained by applying a standard risk-neutral pricing principle.
As to Guaranteed Minimum Accumulation Benefits (GMABs), Mahayni & Schnei-der (2012) conduct a simulation study to investigate whether the utility gained by incorporating a GMAB in a variable annuity can compensate for the loss caused by the upper price setting due to the worst case strategy. Besides, Krayzler, Zagst & Brunner
(2016) provide analytic formulas in a hybrid model for the GMAB and the GMDB with common option riders for actuarial and financial risks.
Guaranteed Minimum Income Benefits (GMIBs), also called Guaranteed Annuity Options (GAOs) in Europe, are discussed in multiple articles. Pelsser (2003) develops a market value for the GAO using martingale modelling techniques and constructs a static replicating portfolio of plain vanilla interest rate swaptions that replicate the GAO. Next, Marshall, Hardy & Saunders (2010) study the valuation of GMIBs in a complete market and examine the sensitivity of the GMIB to financial variables. They find that the fee rates for the GMIB charged by insurance companies is possibly too low. In the article of van Haastrecht, Plat & Pelsser (2010) explicit expressions for GAOs are derived under the assumption of stochastic volatility for the equity prices. They conclude that the impact of ignoring stochastic volatility in valuing GAOs can be sig-nificant. Furthermore,Deelstra & Ray´ee(2013) study the case of valuing GMIBs under local volatility, stochastic volatility and constant volatility models and state that an appropriate volatility model is important for GMIBs. Lastly,Gao, Mamon & Liu(2015) develop a Markov-modulated framework for dependent risk factors for the valuation of GAOs and show that there are significant differences in standard errors and computing times compared to Monte Carlo simulation methods.
Most articles focus on Guaranteed Minimum Withdrawal Benefits (GMWBs) as they are the most common form of a variable annuity. Milevsky & Salisbury(2006) develop a variety of methods for pricing and calculating the costs of GMWBs and conclude with the result that GMWB fees charged in the market are not sustainable and too low. Furthermore, Chen, Vetzal & Forsyth (2008) study the no-arbitrage fee for GMWBs and conclude that the GMWB fees in the market are not enough to cover the cost of hedging the option rider.Dai, Kwok & Zong(2008) develop a singular stochastic control model for pricing variable annuities with GMWBs and explore the optimal withdrawal strategy of a rational policyholder that maximizes the expected discounted value of the cash flows belonging to GMWBs. Next, in the article of Kolkiewicz & Liu (2012) a method of constructing semi-static hedging strategies for GMWBs is developed, which offers certain advantages over dynamic hedging.Kling et al. (2014) analyse the impact of policyholder behaviour on pricing and hedging variable annuities with GMWBs for life using different product designs, market models and approaches for modelling poli-cyholder behaviour. AlsoFung et al.(2014) perform an analysis of how different sets of financial and demographic parameters affect the fair fee charged for a GMWB for life and the profit and loss distribution using stochastic mortality models. Lastly,Bacinello, Millossovich & Montealegre(2016) develop a dynamic programming algorithm for pric-ing variable annuities with GMWBs under a general L´evy process framework.
3.1.2 Unifying valuation methods
Next to the single option rider valuation methods also unifying valuation methods exist.
Hardy(2003) introduces a method to value variable annuities with GMDBs, GMABs and combinations of GMDBs and GMABs.Bauer et al.(2008) develop a unifying framework for valuing variable annuities and Bacinello et al.(2011) propose a unifying framework for the valuation of the guarantees under optimal policyholder behaviour. This subsec-tion provides an overview of the framework presented in the last two articles.
Bauer et al.(2008) start setting up a framework by considering a single premium payed by the policyholder at time zero and a finite integer maturity T of the contract. They set up three different accounts: 1) a general account, 2) a withdrawal account, and 3) a death benefit account. Furthermore, they create benefits bases for all option riders and define a continuous option rider fee which is charged on the general account value.
During the term of the contract, they consider the development of the account values and benefits base values. Each policy year is split up into two parts: 1) from t+ to (t + 1)−, and 2) from (t + 1)− to (t + 1)+ for t = 0, 1, ..., T − 1. From t+ to (t + 1)− the development within the policy year is described. Here the general account value changes due to changes in the price of the underlying mutual fund and due to the withdrawal of the option rider fees. From (t + 1)− to (t + 1)+ the policy anniversary date takes place, at which four different cases can be distinguished: 1) the policyholder has passed away in year (t, t + 1], 2) the policyholder has survived year (t, t + 1] and does not take any action at time t + 1, 3) the policyholder has survived year (t, t + 1] and withdraws an amount within the limits of the GMWB option rider at time t + 1, and 4) the policyholder has survived year (t, t + 1] and withdraws an amount exceeding the limits of the GMWB option rider at time t + 1. In each case changes in the account values and the benefits base values up to time T are calculated according to the formulas described in the article.
After maturity of the contract at time T the living benefit payments are calculated taking into account the used option riders. The contract is valued under a risk-neutral probability measure and under the assumption that the financial markets and biomet-ric events are independent from each other. Using deterministic mortality and survival probabilities, valuation is performed using different types of policyholder behaviour for the withdrawal strategy, namely deterministic, probabilistic and stochastic behaviour.
Bacinello et al. (2011) start with describing the types of approaches the policyholder can use: 1) the static approach, 2) the dynamic approach, and 3) the mixed approach. In the static or passive approach the policyholder does not withdraw cash amounts dur-ing the accumulation period of any option rider or durdur-ing the withdrawal period of a GMIB, withdraws exactly the specified amounts in case of GMWB and never surrenders the contract. The dynamic or active approach assumes that the policyholder withdraws cash amounts which might not coincide with the contractually specified amounts in case of a GMWB and that the policyholder can decide not to withdraw or to surrender the contract at all times. The mixed approach combines the previous approaches and assumes that a policyholder is semi-active. This means that the policyholder withdraws the contractually specified amounts in case of a GMWB and does not withdraw cash amounts during the accumulation period or during the withdrawal period of a GMIB, but the policyholder is allowed to surrender the contract at any time.
pro-posed byBacinello et al.(2010).Bacinello et al.(2011) consider a single premium payed by the policyholder at time zero and a finite integer maturity T of the contract. Firstly, in each policyholder approach the cumulated survival benefits are calculated for each living option rider. The death benefit is calculated thereafter as the maximum of the investment account value and the benefits base of the GMDB. In case of present liv-ing option riders, the death benefit needs to be adjusted when withdrawals or benefit payments are payed before the death of the policyholder. Next, starting with the in-vestment account value equal to the single premium, the evolution of the contract is worked out. Introducing a proportional fee rate and using the return on the investment account value as that of the reference fund, the total cumulated benefits are calculated for each time point and discounted with a money market account based on the risk-free interest rate.
3.2
Valuation over time
Under reporting and controlling guidelines such as the Own Risk and Solvency Assess-ment (ORSA) of Solvency II it is necessary to value variable annuities over time. The simplest method to do so is a nested Monte Carlo simulation method, which is described in Subsection 3.2.1. However, the problem with this method is that it is very time con-suming and computer intensive (Milliman,2013). Therefore, proxy methods have been proposed to value variable annuities over time. The different methods include replicating portfolios, binomial or multinomial trees (see for example Yang & Dai(2013)), partial differential equations and free boundary problems and least-squares Monte Carlo sim-ulation. As the least-squares Monte Carlo simulation method is the most commonly known proxy method, this method is elaborated on in Subsection 3.2.2. Furthermore, the Option Interpolation Model of Ortec Finance can also be used to estimate the value of variable annuities if a general framework to calculate the value is available. The Option Interpolation Model is therefore described in Subsection 3.2.3.
3.2.1 Nested Monte Carlo simulation
The nested Monte Carlo simulation method is the simplest method to value variable annuities over time. The method determines the probability distribution of losses of an insurance company resulting from mismatching assets and liabilities due to adopted assumptions. This probability distribution is then used to calculate the required value of the option riders embedded in variable annuities (Gan & Lin,2015). The method is a two-level procedure: firstly, a Monte Carlo simulation is executed to obtain possible real-world scenarios in which the variable annuity must be valued and, secondly, for each point in time of each real-world scenario a risk-neutral Monte Carlo simulation is executed to obtain the value of the variable annuity at that particular point in time (Cathcart & Morrison,2009;Milliman,2013). By the use of risk-neutral scenarios it is assumed that all agents are risk-neutral when valuing the variable annuity (Hull,2011, Chapter 12). As a variable annuity is considered as an exotic or path-depended option based on the notional amount determined in the contract, the underlying risk prefer-ences of the agents are unimportant and the right price for the variable annuity can be obtained in all worlds. When using risk-neutral scenarios the value of the variable annuity is exactly equal to the discounted expected value of the future payoffs of the contract. Furthermore, risk-neutral valuation of the variable annuity at different points in time implies due to the fundamental theorem of asset pricing that there is no
arbi-trage, so that the law of one price holds for the contract values and the valuation of the variable annuity is generally applicable.
The main disadvantage of the nested Monte Carlo simulation method is that the computation is highly intensive and very time consuming. For example, a single-year projection for one variable annuity contract using 1,000 outer scenarios and 1,000 inner scenarios has already 1,000,000 scenarios in total. For each additional valuation year and each additional contract this number increases strongly.
Reducing the computation time of the nested Monte Carlo simulation method for a large portfolio of variable annuities can be done in several ways (Gan & Lin,2015). First of all, the number of variable annuity contracts can be reduced. This is however not useful as an insurance company is obliged to value all variable annuity contracts it has concluded. Secondly, the number of simulations in the outer loop scenarios can be reduced. Last, the number of simulations in the inner loop scenarios can be reduced. Both options save computing time but at the same time reduce the accuracy of the method. As none of the solutions solve the problem of the method being highly computer intensive and time consuming without creating an accuracy or another unacceptable problem, proxy methods have been developed.
3.2.2 Least-squares Monte Carlo simulation method
The most commonly known proxy method to value variable annuities is the least-squares Monte Carlo simulation method. According toMilliman(2013) the main objective of the least-squares Monte Carlo simulation method is to determine the functional relationship between the risk drivers used for valuation of the variable annuity. Changes in the value of these risk drivers can significantly change the value of the variable annuity.
The method consists of three main steps. First, a large set of real-world Monte Carlo simulations is created which are called the outer scenarios. At a specific point in time, risk-neutral Monte Carlo simulations, also called the inner scenarios, are performed for each real-world Monte Carlo simulation. Normally, the number of inner scenarios is small with a maximum of around ten simulations (Milliman, 2013). For all outer scenarios the value of the variable annuity for the specified time point is calculated using the inner scenarios. As the number of inner scenarios is small, the values will be rather inaccurate. The second step then performs a least-squares regression of the risk drivers on the inaccurate values. In this least-squares regression a linear model is fitted and the coefficients of the risk drivers are found by minimising the sum of squared residuals. Each single inner scenario will provide a poor estimate of the value but regressing over a lot of these inner scenarios will provide stable and converging estimates. The final step calculates the expectation of the value of the variable annuity as the mean of the predicted values given by the fitted model.
The main advantage of the least-squares Monte Carlo simulation method is that the computational task is easily doable (Cathcart & Morrison,2009). Using 1,000 outer scenarios and five inner scenarios only 5,000 simulations are needed. Compared with the nested Monte Carlo simulation method the number of simulations can be reduced by factor twenty in this example. Furthermore, the method is easily implemented as only a simple least-squares regression is used (Longstaff & Schwartz,2001). Other advantages are that it can also capture non-market risk drivers such as mortality rates and lapse assumptions and that the convergence can be proved mathematically. Last, the method is generic enough to have both a risk-neutral as well as a real-world application.
2013). One of the most significant assumptions underlying the method is that the error term is normally distributed. For values that are bounded from below or above, this assumption can be problematic as the resulting distribution will not be symmetric. Furthermore, if the underlying model displays severe discontinuities, the method will not be able to approximate the values properly, resulting in a loss of accuracy.
3.2.3 Option Interpolation Model
The Option Interpolation Model developed by Ortec Finance aims to estimate the value of financial products (Ortec Finance,nd;Singor et al.,2016). The method uses a Monte Carlo simulation to generate real-world scenarios. Then, instead of performing a nested Monte Carlo simulation, an interpolation method is used to determine the value of the financial product in each point in time of all real-world scenarios. The model interpolates on a grid of values of the product that have been calculated beforehand using a risk-neutral Monte Carlo simulation, where the values in the grid should represent all possible values of the product in each point in time in each real-world simulation.
According to Singor et al. (2016) four steps can be identified when approximating the value of the financial product. Firstly, the value of the financial product is analysed to determine the relevant risk drivers. These risk drivers can be economic, such as the interest rate and equity return rate, or non-economic, such as the mortality rate and policyholder behaviour. In the next step the empirical probability distribution function of the risk drivers is identified. In this step a set of real-world scenarios is simulated. Third, the fitting function is specified and the method is calibrated and validated for error assessment. Finally, the method is used for density forecasting in management applications and a realisation of the risk drivers is used to approximate the value of the financial product. This thesis focuses on the first step of this progressive scheme.
Just as the least-squares Monte Carlo simulation method reduces the computational task, the Option Interpolation Model does the same. Furthermore, the method can sup-port a large number of risk drivers, including non-market risks (Ortec Finance,nd). Also prior knowledge of the sensitivities regarding the risk drivers can be incorporated in the model. Last, as the number of interpolation points increases the method will converges to a full nested Monte Carlo simulation method and provides results of the same quality. Comparing the Option Interpolation Model with the least-squares Monte Carlo sim-ulation method, the main difference between both methods lies in the approach of using the risk-neutral and real-world Monte Carlo simulations. The least-squares Monte Carlo simulation method uses a large amount of real-world scenarios and in each real-world scenario only a few risk-neutral scenarios. On the other hand, the Option Interpolation Model first simulates real-world scenarios, then filters a predefined number of inter-polation points out of the real-world scenarios and finally obtains a large number of risk-neutral scenarios for each interpolation point.
The main advantage of the Option Interpolation Model over the least-squares Monte Carlo simulation method is that more accurate values of the financial product are used in the method which results in more stable outcomes (Ortec Finance,nd). Also for tail modelling the Option Interpolation Model is more suitable, as the least-squares Monte Carlo simulation method has to sample a lot of regression points to obtain accurate results in the distribution tail. Last, it is easier to incorporate prior knowledge with respect to the sensitivities of the risk drivers and the tail behaviour in the Option Interpolation Model than in the least-squares Monte Carlo simulation method.
Chapter 4
Methodology
The previous chapters have described the theoretical background of variable annuities and the valuation methods associated with them. Continuing this thesis, this chapter de-scribes the methodology used in the empirical research. The empirical research assumes a single-life variable annuity contract between a policyholder and a Dutch insurance company based on the life of the policyholder which has come into force on January 1, 2017. Firstly, Section4.1describes the specifications in the contract prior to concluding the contract. Next, Section 4.2 discusses the assumptions on the types of policyholder behaviour considered. Section 4.3 focuses on the financial market underlying the con-tract. As the contract takes effect, the modelling framework is described in Section 4.4 and the evolution of the contract is detailed in Section4.5. Finally, Section4.6describes how the present value of the contract can be determined.
4.1
Specifications in the contract
Prior to concluding the contract the contract specifications must be agreed on by the policyholder and the insurance company. Both the policyholder and the insurance com-pany must make decisions regarding the contract. These decisions are clarified here to be able to provide a foundation for the calculations in the following sections. A short overview of the decisions can be found in Table4.1. Subsection 4.1.1elaborates on the decisions of the policyholder, while the decisions of the insurance company are detailed in Subsection4.1.2.
There are two characteristics of the policyholder both the policyholder and the insurance company cannot decide on but which play a crucial role in the contract spec-ification. These characteristics are the current age and the gender of the policyholder. The gender of the policyholder is mainly important for the mortality rates the insurance company decides to use in the contract. The current age of the policyholder has a direct influence on the duration of the contract and indirectly on the mortality rates. These two characteristics form the basis of every contract and are always described.
4.1.1 Decisions of the policyholder
The first decision of the policyholder is regarding the premiums he has to make in order to receive any benefits. The policyholder decides on the initial benefits base, meaning the initial guaranteed amount defined in the contract, and on the number of premium payments he wants to make. The amount of premium to be paid follows as the division of the initial benefits base by the number of premium payments.
Decisions of the policyholder Decisions of the insurance company - Initial benefits base - General fee percentage
- Number of premium payments - Option rider fee percentage
- The use of option riders - Withdrawal percentages for GMWB - Duration accumulation period - Mortality rates
- Duration withdrawal period - Interest rates - Growth of benefits base
- Asset mix
Table 4.1: Short overview of the decisions of the policyholder and the insurance company.
of. The policyholder can choose one of the option riders or make a combination of the Guaranteed Minimum Death Benefit (GMDB) and one of the Guaranteed Minimum Living Benefits (GMLBs). Depending on the option rider(s) chosen the policyholder has to make additional decisions.
The next decision is about the duration of the contract. The duration of the contract is influenced by the option rider(s) the policyholder has chosen. Choosing the GMDB or the Guaranteed Minimum Accumulation Benefit (GMAB) the policyholder has to decide on the age of termination of the contract, while choosing the Guaranteed Minimum Income Benefit (GMIB) or the Guaranteed Minimum Withdrawal Benefit (GMWB) the policyholder faces two duration issues. First, the policyholder has to decide on the starting age of the benefit payments. Second, the policyholder has to decide on the period of time he wants to receive the benefit payments where he has two options: a whole-life benefit period or a temporary benefit period. The whole-life benefit period provides benefit payments for the entire remaining life of the policyholder, while the temporary benefit period provides benefit payments for a predefined period of time. For the temporary benefit period an ending age of the benefit payments must be chosen.
Fourthly, the policyholder has to decide on the future growth of the benefits base. For each option rider the policyholder has three options for the growth factor: the return of premiums, the roll-up and the ratchet. When opting for the roll-up, the policyholder has to decide on the roll-up percentage. As described in the literature study there is an additional growth factor, the reset, which is not considered in this thesis due to time and modelling constraints.
Last, the policyholder has to decide on the asset mix in the investment account. To constrain calculation time, the policyholder can choose a mix of a bank account, earning the risk-free interest rate, and stocks, earning a more risky rate. This thesis assumes for simplicity that the policyholder does not change his asset mix during the time the contract is in force.
Naturally, the policyholder is not entirely free in making his decisions. Regarding the growth factors of the benefits base for example, the insurance company sets limits on the roll-up percentage. Furthermore, the insurance company has directives for the asset mix or a maximum for the percentage invested in stocks. In reality, an insurance company does not offer all possible contracts which also restricts the policyholder. 4.1.2 Decisions of the insurance company
After the decisions of the policyholder have become known to the insurance company, the insurance company has to make decisions based on the policyholder’s information. Regarding the fees, the insurance company has two main sources of costs for which fees must be paid by the policyholder. First, the insurance company faces general costs.
These costs cover all administrative expenses the insurance company has to make to be able to provide the contract. The insurance company asks a fixed percentage of the investment account each year the contract is in force as a general fee to cover these costs. Second, the insurance company makes costs to provide for the option riders the policyholder has chosen. For each option rider the insurance company charges an option rider fee, which is calculated as a fixed percentage of the investment account. However, the duration of this fee depends on the chosen option rider. For the GMDB and GMAB the fee is charged during the entire time the contract is in force, while for the GMIB and GMWB the fee is only charged during the accumulation period.
Next, the insurance company determines the withdrawal percentages in the GMWB. If the policyholder has chosen a whole-life benefit period, the insurer sets a fixed with-drawal percentage in the range of four to seven per cent depending on the starting age of the benefit payments. If the policyholder has chosen a temporary benefit period, the withdrawal percentage is set equal to the benefits base at the end of the accumulation period divided by the number of years the policyholder receives the benefit payments. However, this percentage is not allowed to be smaller than the withdrawal percentage in the whole-life benefit period with the same starting age of the benefit payments.
Last, the insurance company has to decide on the assumptions underlying the con-tract, for example the mortality rates. To be able to provide a reliable estimate of the contract value it is best to use the most recent mortality table. Often, as the policy-holders do not reflect the average population on which the mortality table is based, the insurance company adapts the mortality rates with policyholder characteristics to obtain mortality experience tables. Another assumption is on the interest rates for cal-culations in and discounting of the contract. Again the most recent interest rates will provide the most reliable estimates of the contact value.
4.2
Policyholder behaviour
Following the article of Bacinello et al. (2011), this thesis assumes different types of policyholder behaviour. WhereBacinello et al.(2011) assume three types of policyholder behaviour, namely static, dynamic and mixed behaviour, this thesis only considers static and mixed policyholder behaviour due to time and modelling constraints.
Static policyholder behaviour assumes that the policyholder does not withdraw any cash amounts from his investment account during the accumulation period of any option rider or during the withdrawal period of the GMIB. Furthermore, if the contract contains a GMWB, this type of behaviour assumes that the policyholder withdraws exactly the cash amounts which are contractually specified at the predefined dates. Last, it is assumed that the contract is never surrendered by the policyholder.
Mixed policyholder behaviour also assumes that the policyholder withdraws exactly the cash amounts contractually specified at the predefined dates in case of a GMWB. Furthermore, the policyholder also does not withdraw any cash amounts from his in-vestment account during the accumulation period of any option rider and during the withdrawal period of the GMIB. However, the policyholder is allowed to surrender the contract at any time during the time the contract is in force.
4.3
The financial market
clari-fied, this thesis regards the financial market the contract is valued in. As this thesis wishes to find a general valuation framework for variable annuities and a variable an-nuity can be considered as an exotic or path-depended option based on the notional amount determined in the contract, no arbitrage, as defined under the fundamental theorem of asset pricing, is assumed. This means that the law of one price holds and the valuation framework is general applicable. A necessary condition of no arbitrage is a risk-neutral valuation framework, in which the value of the variable annuity is exactly equal to the discounted expected value of the future payoffs of the contract. Therefore, a risk-neutral world is assumed in this thesis. In this risk-neutral world, the financial market must be able to produce interest rates as well as bank account values and stock prices. The two-factor Hull-White Black-Scholes model is implemented to model the financial market. Several steps have to be taken. First of all, the Nelson-Siegel formula is applied to a recent term structure. Next, the parameters of the two-factor Hull-White model are obtained. Last, interest rates, bank account values and stock prices are cre-ated using the two-factor Hull-White Black-Scholes model.
The first step is to apply the Nelson-Siegel formula to a recent term structure. The term structure used in this thesis is the swap curve of December 30, 2016, provided by Ortec Finance through its data provider. The swap curve is chosen as it is the most liquid term structure and therefore contains the most reliable data to base the financial market on. The swap curve consists of swap rates with a maturity up to hundred years. The parameters of the Nelson-Siegel formula, developed inNelson & Siegel(1987) and given by the formula
r(t) = β0+ β1 1 − exp −τt t τ + β2 1 − exp −τt t τ − exp −t τ ! = β0+ (β1+ β2) 1 − exp −t τ τ t − β2exp −t τ ,
are fitted on the swap curve. Here, t is the maturity and r(t) is the interest rate at maturity t. Parameter β0 is the long-term level of the interest rate curve which is a
constant, while parameter β1 is the short-term level describing the slope of the curve.
Parameter β2 is the medium-term level of the interest rate curve and describes the
cur-vature of the curve. Furthermore, τ is the decay factor which describes the convexity of the curve. The fitted parameter values for the swap curve can be found in the first column of Table 4.2.
The attention now shifts to the two-factor Hull-White model. The Hull-White model is a stochastic model that describes the instantaneous short rate and uses the most recent term structure as input for consistency. The two-factor Hull-White model is preferred over the one-factor Hull-White model as the second factor describes the volatility and correlation of the interest rate more precisely. Including the second factor makes the variability of the interest rates more market consistent and introduces a non-perfect correlation between interest rates of different maturities. The two-factor Hull-White model under the risk-neutral measure is given by (Brigo & Mercurio,2006, Section 4.2):
dr(t) = (θ(t) + u(t) − ar(t)) dt + σrdZr(t), r(0) = r0,
du(t) = −bu(t)dt + σudZu(t), u(0) = 0.
Here, (Zr(t), Zu(t)) is a two-dimensional Wiener process with dZr(t)dZu(t) = ρrudt.
