On the impact of stochastic volatility, interest rates and mortality on the hedge efficiency of GLWB quarantees

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On the impact of stochastic volatility, interest rates and

mortality on the hedge efficiency of GLWB guarantees

Mémoire Pierre-Alexandre Veilleux Maîtrise en actuariat Maître ès sciences (M.Sc.) Québec, Canada © Pierre-Alexandre Veilleux, 2016

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Résumé

Les rentes variables, et plus particulièrement les garanties de rachat viager (GRV), sont deve-nues très importantes dans l’industrie de la gestion du patrimoine. Ces garanties, qui offrent aux clients une protection de revenu tout en leur permettant de garder une participation dans les marchés boursiers, comprennent différents risques systématiques du point de vue de l’émetteur. La gestion des risques des GRV est donc une préoccupation majeure pour les compagnies d’assurance, qui ont opté pour la couverture sur les marchés financiers comme stratégie de gestion des risques simple et efficace.

Ce mémoire évalue l’impact de la modélisation du passif de la garantie sur l’efficacité de la couverture des GRV par rapport à trois risques systématiques importants pour ces garanties, soient les risques de marchés boursiers, d’intérêt et de longévité. Le présent travail vise donc à étendre l’analyse effectuée par Kling et al.(2011), qui se concentre sur le risque de marchés boursiers. Ce mémoire montre que les taux d’intérêt stochastiques sont primordiaux dans la modélisation du passif des GRV.

Ce mémoire analyse également l’impact de la modélisation de la mortalité utilisée dans la boucle externe sur l’efficacité de la couverture des GRV. Une allocation du risque entre les risques financiers et le risque de longévité est utilisée pour montrer que la longévité représente une part importante du risque total des GRV couvertes. De plus, l’efficacité de la couver-ture dans des projections incluant une modélisation stochastique des risques financiers et du risque de longévité est comparée à l’efficacité dans des projections utilisant des marges pour écarts défavorables traditionnelles sur l’hypothèse d’amélioration de mortalité. La diversifica-tion entre les risques financiers et de longévité s’avère avoir un effet substantiel sur l’efficacité de la couverture.

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Abstract

Variable annuity guarantees, and particularly guaranteed lifetime withdrawal benefit (GLWB) guarantees, have become very important in the wealth management industry. These guaran-tees, which provide clients with a revenue protection while allowing them to retain equity market participation, exhibit significant systematic risks from the issuer’s standpoint. Risk management of GLWB guarantees thus is a main concern for insurance companies, which have turned to capital market hedging as a straightforward and effective risk management method. This thesis assesses the impact of the guarantee liability modeling on the hedge efficiency of GLWB guarantees with respect to three significant systematic risks for these guarantees, namely, the stock market, interest rate and longevity risks. The present work thus aims to extend the hedge efficiency analysis performed inKling et al.(2011), which focuses on the stock market risk. In this thesis, stochastic interest rates are shown to be of primary importance in the guarantee liability modeling of GLWB guarantees.

This thesis also analyzes the impact of the outer loop modeling of mortality on the hedge efficiency of GLWB guarantees. A risk allocation between financial and longevity risks is used to show that longevity holds a significant share of the total risk of a hedged GLWB guarantee. The hedge efficiency in projections including both stochastic financial and mortality modeling is compared with the efficiency in projections using traditional actuarial margins for adverse deviations on the mortality improvement assumption. The diversification between financial and longevity risks is shown to have a substantial impact on hedge efficiency.

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List of Tables

1.1 Living benefit guarantees features comparison . . . 4

2.1 Illustration of a risk-neutral scenario used in the valuation of the GLWB guar-antee liability at time 0 . . . 26

5.1 Parameters for the regime-switching lognormal model . . . 58

5.2 Quantiles of the stock index with SP 0 = 100 . . . 59

5.3 Parameters of the two-factor extended Vasicek model . . . 60

5.4 Parameters of the Lee-Carter model . . . 61

5.5 Contract holder parameters . . . 62

5.6 Contractual parameters . . . 63

5.7 Lifetime withdrawal rates by attained age . . . 63

5.8 Average tail hedged and unhedged losses (as a % of SP ) . . . 72

5.9 Impact of the hedging strategy on hedge efficiency . . . 73

6.1 Parameter sets for the Hull-White model. . . 77

6.2 One-year spot rate movement volatilities . . . 77

6.3 Impact of the guarantee liability modeling on the initial guarantee liability. . . 80

6.4 Impact of the guarantee liability modeling on hedge efficiency for the 65-year-old contract holder . . . 81

6.5 Impact of the guarantee liability modeling on hedge efficiency for the 50-year-old contract holder . . . 84

7.1 Impact of mortality deviations on hedge efficiency for the 65-year-old contract holder using a 3-rho hedging strategy. . . 88

7.2 Impact of mortality deviations on average tail losses for the 65-year-old contract holder without hedging. . . 89

7.5 Allocation of risk with a 3-rho hedging strategy for the 65-year-old contract holder . . . 92

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7.8 Extended risk allocation with a 3-rho hedging strategy for the 65-year-old

con-tract holder . . . 95

7.9 Extended risk allocation with a 5-rho hedging strategy for the 65-year-old

con-tract holder . . . 95

7.10 Assessment of the diversification between financial and longevity risks for the

65-year-old contract holder using a 3-rho hedging strategy . . . 97

65-year-old contract holder using a 5-rho hedging strategy . . . 99

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List of Figures

1.1 LIMRA’s annuity sales estimates . . . 2

1.2 GLWB guarantee illustration . . . 6

1.3 Illustration of a triennial ratchet (left panel) and a yearly roll-up (right panel)

in the accumulation phase of a GLWB guarantee . . . 8

3.1 Key-rate rho shocks on the spot curve . . . 35

5.1 Quantiles of the 1-year (left panel) and 30-year (right panel) spot rates in the

two-factor extended Vasicek model . . . 60

5.2 Estimated αx (left panel) and βx (right panel) values by age in the Lee-Carter

model . . . 62

5.3 Quantiles of the lifetime withdrawal amount (left panel) and distribution of the

time until the account value is exhausted (right panel) . . . 64

5.4 Unhedged GLWB guarantee gains and losses distribution. . . 65

5.5 Smoothed distributions of the hedged (P V(k)

H ) and unhedged (P V (k) U + V

P,(k) 0 )

GLWB guarantee gains and losses. . . 72

6.1 Average paths of VP ti − V

P,S−

ti (left panel) and of the stock market index (right

panel) . . . 82

7.1 Quantiles of the future path of κtused as assumptions with margins for adverse

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Remerciements

Je voudrais d’abord remercier mon superviseur, M. Etienne Marceau, pour m’avoir accom-pagné dans ce projet qui m’a été cher. M. Marceau m’a à la fois grandement aidé dans la rédaction de ce mémoire, m’a offert son support moral dans les moments de doute, et m’a offert la remarquable opportunité de présenter les résultats de mes travaux lors de colloques et ainsi de pouvoir échanger avec différents collègues des milieux académique et pratique. En somme, M. Marceau a fait de mes études aux cycles supérieurs une expérience très enrichissante. Je me dois également de remercier les évaluateurs de mon mémoire, M. David Landriault et M. Patrice Gaillardetz, qui m’ont donné de judicieux commentaires sur ce travail.

J’aimerais remercier du fond du cœur les personnes qui ont gravité autour de moi au cours de ces quelques dernières années. Tout d’abord, mon épouse, Marie-Eve, pour qui je décrocherais volontiers la lune, et aussi ma mère, Doris, la plus merveilleuse des mamans. Ces deux femmes ont une générosité envers moi qui n’a pas d’égal, et je leur en serai toujours reconnaissant. Un immense merci également à mon père et à ma sœur, sur qui je peux toujours compter et qui m’ont encouragé à travers ce projet. Merci également à toutes ces autres personnes que je côtoie et que j’apprécie, et auxquelles je n’ai pas pu consacrer le temps que j’aurais voulu. J’aimerais également remercier les personnes qui ont rendu ce projet possible en me permettant de conjuguer travail et étude, soient M. Mario Robitaille, M. Frédéric Tremblay et Mme Kim Girard. Un merci particulier à mon collègue Maxime Turgeon-Rhéaume, qui a su me motiver à la fois à entreprendre ma maîtrise et à la poursuivre malgré l’engagement de temps que cela pouvait représenter.

Enfin, j’aimerais exprimer ma gratitude à l’Industrielle Alliance pour son soutien financier en lien avec les coûts liés à la poursuite de mes études à la maîtrise. J’aimerais aussi remercier la Chaire d’actuariat de l’Université Laval et le Fonds de soutien à la réussite de l’Université Laval pour leur soutien financier.

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Chapter 1

Introduction

1.1 Variable annuities

Wealth management and retirement planning have been growing concerns in the general pop-ulation as poppop-ulation ageing has pushed more people into the accumpop-ulation and retirement phases of their lives. Insurance companies and other players in the wealth management indus-try have responded to the wealth management and retirement planning needs with a full breath of products for potential clients. These products include mutual funds, annuities, guaranteed interest contracts, bonds, variable annuities and other similar products.

Variable annuities are a relatively recent development in the wealth management industry. Indeed, although they were first introduced in 1952 in the United States, they have gained much of their popularity in the 1990s (Poterba (1997)), and new product designs are still emerging on the market.

Basic variable annuity contracts are often seen as being similar to mutual fund investments: these two investment vehicles allow the accumulation of money in various investment options, with little constraints regarding the timing of additional premiums or withdrawals from the contract. Moreover, the contract holder is at risk regarding the investment performance, which depends on the chosen investment options.

However, variable annuity contracts also have characteristics that set them apart from mutual fund investments. First of all, variable annuities allow the accumulation of tax-deferred sav-ings, which may not always be the case for mutual funds. For variable annuities, income and investment gains are only taxed as money is withdrawn from the funds. Moreover, and often most importantly, variable annuities offer guarantees to contract holders (U.S. Securities and Exchange Commission(2011)). These guarantees provide contract holders with some kind of protection on their investment, thus reducing financial risks compared with traditional mutual fund investments. Finally, variable annuity guarantees are sold by insurance companies only.

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Although variable annuities share their name with traditional annuities, these two products have significant dissimilarities. Indeed, variable annuities offer more flexibility than traditional annuities in certain respects. First, they offer a more flexible range of investment options than traditional annuity contracts. Investment options include money market funds, bond funds, stock market funds, diversified funds and more. Moreover, variable annuities allow contract holders to maintain their liquidity, meaning that they can withdraw their funds if they wish to lapse the variable annuity contract.

Variable annuities have attracted significant interest for several years now, even outperforming fixed annuities in sales. Figure 1.1 shows LIMRA’s annuity sales estimates since 2001 (Liu

(2010) and LIMRA Secure Retirement Institute (2014)). LIMRA, formerly known as the Life Insurance and Market Research Association, is an insurance and financial service trade association that provides research, learning and development programs to financial services companies around the world.

Figure 1.1: LIMRA’s annuity sales estimates

Variable annuity sales are shown in Figure1.1to be consistently high for an extended period of time. Macroeconomic conditions sure are an important cause for the variable annuities’ popularity. With lowering fixed annuity payouts due to steadily decreasing interest rates over the past three decades, variable annuities became an attractive alternative for retirement. Indeed, they allow contract holders to accumulate money while avoiding locking the current low payout rates. Moreover, as will be shown in section1.2, some variable annuity guarantees are dedicated to providing a retirement income.

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Compensation may be another factor for the variable annuities’ popularity, as these products often offer more generous commissions than mutual funds or traditional annuities.

Finally, guarantees offered in variable annuities are no doubt another explanation for their popularity, as they allow contract holders to have both a participation in equity markets and the protection provided by the guarantees. They are thus suited to risk-averse investors. Nowadays, variable annuities are at the heart of the wealth management, retirement plan-ning and estate planplan-ning business in the United States. In addition to being popular in the United States, variable annuities have counterparts in other countries. In Canada, segregated funds have much similarities with variable annuities. In Europe, the term “investment-linked product” is used to denote products that include guarantees linked to market performance. In all cases, the guarantees offered in these products set them apart from traditional invest-ments by providing protection to contract holders. Typical variable annuity guarantees are described in more details in section 1.2.

1.2 Variable annuity guarantees

The common goal of all variable annuity guarantees is to provide protection on the investment made by a contract holder in a variable annuity contract. Typical variable annuity guarantees can be classified in two broad categories: death benefit guarantees and living benefit guaran-tees. Death benefit guarantees only include the guaranteed minimum death benefit (GMDB) guarantee, which provides a guaranteed value to the contract holder at the time of death. This implies that no matter the account value of the contract holder at the time of death, the insurance company pays the maximum between the account value and the guaranteed value to the contract holder’s estate. The guaranteed value usually is a given percentage of the initial and subsequent investments in the variable annuity contract.

Living benefit guarantees include a variety of guarantees. First of all, the guaranteed minimum accumulation benefit (GMAB) guarantee provides the contract holder with a guaranteed value at maturity of the contract. Thus, the company pays the maximum between the account value and the guaranteed value at a specified maturity date. The guaranteed value is often a percentage of the initial and subsequent investments in the variable annuity contract. The guarantee may or may not be renewable.

The guaranteed minimum income benefit (GMIB) guarantee provides the contract holder with a minimum annuity payout. At the time of annuitization, the contract holder is entitled to an annuity whose payout is the maximum between the guaranteed minimum annuity payout and the annuity payout determined using the account value and the annuity rate at the time of annuitization. The guarantee is only applicable if the contract holder annuitizes. Contract holders lose their liquidity after annuitization, as no withdrawals in excess of the annuity

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payout can be made.

The guaranteed minimum withdrawal benefit (GMWB) guarantee provides the contract holder with a yearly guaranteed withdrawal amount over a given time period. A significant difference between GMWB guarantees and GMIB guarantees is that for GMWB guarantees, contract holders remain invested in the various investment options offered in variable annuities while withdrawing. Contract holders thus keep their liquidity at all time in a GMWB guarantee. If guaranteed withdrawals exhaust the contract holder’s account value, the contract holder is still entitled to receive the guaranteed withdrawal amount for the remainder of the guarantee period. Moreover, any remaining account value at maturity of the guarantee or at death of the contract holder is returned to the contract holder or to her estate.

The guaranteed lifetime withdrawal benefit (GLWB) guarantee is similar to the GMWB guar-antee. It provides the contract holder with a lifetime guaranteed withdrawal amount while funds remain invested in the various investment options offered in variable annuities. Once again, the company is liable to pay guaranteed withdrawal amounts once the contract holder’s account value is exhausted. Any remaining account value at death is paid to the contract holder’s estate. The main difference between GLWB and GMWB guarantees is thus the life-contingent nature of GLWB guarantees.

The important features of living benefit guarantees are compared in Table 1.1. First of all, the GMAB guarantee is the only one that does not provide a steady income, making it less attractive as a retirement product. Moreover, the GMIB guarantee is the only guarantee in which liquidity has to be forsaken, thus providing less flexibility to retirees. Finally, among the two guarantees left, the GLWB guarantee is the only one providing a lifetime income, making it particularly attractive as a retirement product.

Feature GMAB GMIB GMWB GLWB Guaranteed income No Yes Yes Yes

Liquidity Yes No Yes Yes

Lifetime No Yes No Yes

Table 1.1: Living benefit guarantees features comparison

GMWB and GLWB guarantees hold an important market share of total variable annuity guar-antees. In the United States, a 2013 Towers Watson survey showed that the GMWB inforce of 14 of the 18 largest variable annuity writers represented 266 billion of dollars (seeTowers Watson(2013)). The same survey revealed that 38% of the variable annuity inforce included a GMWB or GLWB guarantee. In Canada, GMWB and GLWB guarantees contributed to gen-erating very high segregated fund sales. In 2007 to 2010 alone, GMWB and GLWB guarantees generated nearly 20 billion in net sales in the Canadian market (seeTheriault(2011)).

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Lately, the decrease in interest rates and high capital requirements forced some insurers, par-ticularly in Canada, to stop offering GMWB and GLWB guarantees or significantly lower the guarantees offered. Nonetheless, insurers currently have large blocks of GMWB and GLWB guarantees to manage. Therefore, the risk management strategies and risk assessments as-sociated with GMWB and GLWB guarantees remain a hot topic in the insurance industry. Indeed, at the time of writing this thesis, the Canadian federal and Quebec’s provincial regu-lators are making progress on defining new capital requirement formulas for segregated fund guarantees, and place much emphasis on GMWB and GLWB guarantees.

GLWB guarantees will be described in more details in section 1.3.

1.3 Guaranteed lifetime withdrawal benefit guarantees

The focus of this thesis being GLWB guarantees, this section aims to describe these guarantees in more details by presenting the relevant technical terminology, describing the guarantee phases and discussing the guarantee features.

1.3.1 Definitions

The description of GLWB guarantees laid out in this section is based on a few technical terms that shall first be defined. First of all, the account value (AV) denotes the current value of the premiums invested in the variable annuity. The account value is impacted by the underlying funds’ returns, contractual fees and withdrawal payments. Notwithstanding guarantees, the account value is the amount that the contract holder is entitled to at any time in the variable annuity contract.

The lifetime withdrawal amount (LWA) is the withdrawal amount that the insurance company guarantees to the contract holder for the entire duration of the GLWB guarantee, that is to say, for the contract holder’s lifetime. It is the maximum withdrawal that a contract holder can make in a given year without incurring penalties on the guarantee.

Finally, the guaranteed withdrawal balance (GWB) is used to determine the lifetime with-drawal amount. The lifetime withwith-drawal amount is a contractual percentage of the guaran-teed withdrawal balance at the time the contract holder starts making withdrawals. Once withdrawals have started, the guaranteed withdrawal balance decreases on a dollar-for-dollar basis with withdrawal payments. The guaranteed withdrawal balance is not affected by the underlying funds’ returns and the contractual fees.

1.3.2 Guarantee phases

There are two phases in GLWB guarantees. The first of these phases is the accumulation phase. In the accumulation phase, no withdrawals are made by the contract holder. Investments in

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the variable annuity contract are rather allowed to evolve with fund returns and fees. The accumulation phase can be of particular interest for contract holders because of the various features that can increase the guaranteed withdrawal balance during this phase. These features are described in section1.3.3.

The second phase of the GLWB guarantee is the withdrawal phase. During the withdrawal phase, periodic withdrawals of the lifetime withdrawal amount are made by the contract holder. As long as the contract holder’s account value is positive, the withdrawals are taken out of the account value. However, if the account value reaches zero because of guaranteed withdrawals and poor market performance, the insurance company must pay the lifetime withdrawal amount to the contract holder.

The two phases of GLWB guarantees are illustrated in Figure1.2. The guarantee is assumed to be sold to a 65-year-old contract holder with a 6% withdrawal rate and to include a 5-year accumulation phase. Figure 1.2 shows that in the accumulation phase, no withdrawals are made and the account value thus fluctuates based on market performance. Then, the contract holder transfers to the withdrawal phase, and lifetime withdrawals start being made. The account value then still fluctuates with market returns, but also decreases as a result of with-drawals. When the account value eventually reaches zero, the company becomes responsible for paying the lifetime withdrawal amount to the contract holder until death.

65 68 71 74 77 80 83 86 89 92 95 98 101 104 Age 0 20 40 60 80 100 120 140 Account value

Lifetime withdrawal amount

Figure 1.2: GLWB guarantee illustration

The GLWB guarantee being an addition to the variable annuity contract, it does not alter the underlying contract. Therefore, at death of the contract holder, the contract holder’s estate is entitled to any remaining account value in the contract.

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1.3.3 Additional features

As mentioned above, GLWB guarantees contain additional features that are meant to increase their competitiveness. These features impact the guaranteed withdrawal balance.

The guaranteed withdrawal balance at inception of the contract is equal to the amount ini-tially invested in the variable annuity. Then, its evolution depends on the guarantee features included in the contract. Popular features in the Canadian and US markets include ratchets and roll-ups of the guaranteed withdrawal balance.

The ratchet feature allows the guaranteed withdrawal balance to be increased to the account value level periodically, for example every three years, if the latter is higher than the former. The ratchet feature thus allows contract holders to participate in strong market performance in the accumulation phase. Ratchets may also be allowed in the withdrawal phase. Then, ratchets increase either the guaranteed withdrawal balance, the lifetime withdrawal amount, or both. The lifetime withdrawal amount may never decrease following a ratchet. GLWB guarantees typically also include an additional automatic ratchet at the time of transfer from the accumulation phase to the withdrawal phase.

The roll-up feature allows the guaranteed withdrawal balance to be increased by a stated percentage as long as no withdrawals are made in a given year. Hence, each year during the accumulation phase, the guaranteed withdrawal balance is increased by the contractual roll-up percentage. This feature encourages contract holders to accumulate money in the GLWB product, since a longer accumulation period implies more roll-ups of the guaranteed withdrawal balance.

Neither of the ratchet feature nor the roll-up feature increase the contract holder’s account value. However, both these features contribute in increasing the guaranteed withdrawal bal-ance at the time of transfer from the accumulation phase to the withdrawal phase, and as such contribute in increasing the lifetime withdrawal amount. The two features described above are illustrated in Figure 1.3.

Figure 1.3 shows how the guaranteed withdrawal balance may evolve in time based on the guarantee features considered. In this figure, the contract holder is 65 years old at inception of the contract. The x-axis must be interpreted as the age of the contract holder as time goes by in the variable annuity contract.

The left panel of Figure 1.3 shows that a triennial ratchet feature increases the guaranteed withdrawal balance to the account value level every three years if the latter is higher than the former. The ratchet feature is thus contingent on a strong market performance. Moreover, the right panel of Figure 1.3 shows that the roll-up feature causes a steady increase in the guaranteed withdrawal balance each year. The roll-up feature is applied regardless of the account value level.

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65 66 67 68 69 70 71 100 110 120 130 Age ● ● ● ● ● ● ● ● AV GWB 65 66 67 68 69 70 71 100 110 120 130 Age ● ● ● ● ● ● ● ● AV GWB

Figure 1.3: Illustration of a triennial ratchet (left panel) and a yearly roll-up (right panel) in the accumulation phase of a GLWB guarantee

1.4 Literature review

This thesis is interested in the risk assessment of GLWB guarantees. Hence, in this section, a review of the relevant literature on GLWB guarantees is presented. Because GMWB and GLWB guarantees are closely related to one another and that there is a larger body of literature on GMWB guarantees, the relevant literature on GMWB guarantees is presented first, followed by the literature on GLWB guarantees.

1.4.1 GMWB guarantees

The problem of pricing and hedging GMWB guarantees has attracted growing interest in the academic literature in the last few years. Milevsky and Salisbury(2006) is to our knowledge one of the first articles dealing with the problem of pricing GMWB guarantees. Under the Black-Scholes framework and assuming a simple GMWB guarantee with continuous withdrawals, they find the fair guarantee charge under the passive and dynamic contract holder behaviour assumptions.

The passive (or static) and the dynamic (or optimal) behaviours are at the ends of the spectrum regarding contract holder sophistication. At one end, a passive contract holder behaviour implies that the contract holder withdraws the lifetime withdrawal amount consistently until death. At the other end, a dynamic contract holder behaviour refers to the strategy in which

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the contract holder makes withdrawals so as to maximize the expected cash flows from the GMWB or GLWB contract. A dynamic behaviour thus essentially becomes a loss-maximizing behaviour from the insurance company’s standpoint.

Under the passive behaviour assumption, Milevsky and Salisbury (2006) shows that the GMWB guarantee can be decomposed into a Quanto Asian put plus a generic term-certain annuity. Under the dynamic behaviour assumption, they show that the valuation problem be-comes an optimal stopping problem similar to pricing an American put option. Their analysis leads them to conclude that GMWB guarantees are underpriced in the market.

Using a framework similar to the one presented in Milevsky and Salisbury (2006), Dai et al.

(2008) develops a singular stochastic control model for pricing GMWB guarantees under the dynamic contract holder behaviour assumption and propose a finite difference algorithm using the penalty approximation approach to solve it. They also extend the valuation to the case of discrete withdrawals.

Whereas Milevsky and Salisbury (2006) and Dai et al. (2008) focus on GMWB guarantees,

Bauer et al.(2008) introduces a model meant to allow the consistent valuation of all variable annuity guarantees. They calculate the fair guarantee charge under the passive and dynamic contract holder behaviour assumptions, using Monte Carlo simulation for the passive behaviour and a generalization of a finite mesh method for the dynamic behaviour. Their model is in discrete time, considers the effect of mortality and allows for yearly ratchets of the guaranteed withdrawal balance.

The pricing of GMWB guarantees under the dynamic contract holder behaviour assumption is also studied in Chen et al. (2008). Their article generalizes some of the assumptions made in e.g. Dai et al. (2008) by introducing a split between mutual fund fees and the guarantee charge and considering a jump diffusion model in addition to the geometric Brownian motion in pricing GMWB guarantees. Both of these generalizations increase the GMWB fair guarantee charge. They also consider the effect of sub-optimal contract holder behaviour on the guarantee value and conclude that GMWB guarantees are underpriced in the market.

The framework ofMilevsky and Salisbury(2006) under the passive contract holder behaviour assumption is extended in Peng et al. (2012), which models interest rates using the Vasicek model. Once again, writing the GMWB guarantee as the combination of a put option and an annuity, they are able to find lower and upper bounds for the GMWB guarantee value by using Roger-Shi’s and Thompson’s approximation methods.

As in Bauer et al. (2008), Banicello et al. (2011) introduces a unifying framework for the valuation of all variable annuity guarantees. Within this framework, they price a GMWB guarantee under the passive approach and the mixed approach using Monte Carlo simulation and least squares Monte Carlo simulation, respectively. The mixed approach is similar to

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the passive approach, but includes the possibility of full surrender at an optimal time. Their article goes beyond the Black-Scholes framework used in Bauer et al. (2008) by introducing stochastic volatility and interest rates through the use of the Heston model and the Vasicek model. They also introduce stochastic mortality in the modeling by the use of a square root process for the force of mortality.

A framework similar to the one presented inMilevsky and Salisbury(2006) is used inLiu(2010) to analyze the pricing and hedging of GMWB guarantees with discrete withdrawals under the passive behaviour assumption. Liu(2010) first prices the GMWB guarantee from the contract holder and from the insurance company’s standpoint and shows that the two approaches are in fact equivalent. Moreover, in addition to investigating dynamic hedging strategies under this framework, she proposes semi-static hedging strategies that improve the hedge efficiency, especially when jumps in prices are taken into account in measuring the efficiency. She also extends the proposed semi-static hedging strategies to the stochastic volatility case using the Heston model.

Similar ideas in terms of pricing and hedging in the Black-Scholes framework are presented in Kolkiewicz and Liu (2012). Once again, a semi-static hedging strategy is proposed and compared to dynamic hedging. The semi-static hedging strategy is shown to outperform the dynamic hedging strategy when there are random jumps in asset prices.

A flexible tree for the valuation of GMWB guarantees is proposed inYang and Dai (2013). Their tree allows them to evaluate the GMWB fair guarantee charge without introducing significant numerical pricing errors, even when the guarantee includes more complex features such as deferred withdrawals, roll-ups and ratchets. Mortality risk and the possibility of full surrender may also be incorporated into the pricing model. Their analysis is done in the Black-Scholes framework.

By re-writing the GMWB pricing problem in the form of an Asian styled claim and a dimensionally-reduced partial differential equation,Donnelly et al.(2014) is able to price a GMWB guarantee under stochastic volatility and stochastic interest rates. They depart from the research work made previously by modeling the fund value has a mix of equities and bonds. They solve the PDE using an Alternating Direction Implicit method.

Banicello et al.(2014) presents a dynamic algorithm for pricing GMWB guarantees under a general Lévy process framework. They consider the geometric Brownian motion, the Merton jump diffusion, the variance-gamma and the Carr, Geman, Madan, Yor (CGMY) models as special cases of the Lévy process modeling. They also compare fair guarantee charges under passive and dynamic contract holder behaviours.

Finally,Feng and Volkmer(2015) develops semi-analytical solutions for pricing GMWB guar-antees both from the insurer’s perspective and the contract holder’s perspective under the

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Black-Scholes framework. They show that assuming no friction costs, the two approaches are equivalent. Their semi-analytic solutions lead to a fast and accurate algorithm for pricing GMWB guarantees.

1.4.2 GLWB guarantees

GLWB guarantees have attracted somewhat less attention in the literature than GMWB guarantees. Nonetheless, there is a growing body of literature on the subject.

Using a continuous-time GLWB guarantee under the Black-Scholes framework and a simpli-fying assumption for population mortality, Shah and Bertsimas(2008) presents an analytical solution for the GLWB fair guarantee charge under the passive contract holder behaviour. They then generalize their modeling to consider stochastic interest rates and volatility through the use of the two-factor Vasicek model and the Heston model, and also allow for an arbi-trary population mortality. They conclude that there is insufficient price discrimination in the market and that model risk can be significant in determining the fair guarantee charge. The pricing of GLWB guarantees under the Black-Scholes framework is also considered in

Piscopo and Haberman (2011), which presents the calculation of the GLWB fair guarantee charge under the passive contract holder behaviour using Monte Carlo simulations. They test the sensitivity of the fair charge to key parameters and give special attention to mortality risk by using a stochastic mortality model. They also evaluate the cost of additional features, such as roll-ups, ratchets and deferred withdrawals.

A different angle is taken by Ngai and Sherris (2011), which investigates the effectiveness of static hedging strategies on longevity risk using longevity bonds and derivatives. They conclude that the hedge is not effective for GLWB guarantees because of its exposure to market risk.

Holz et al.(2012) extends the framework presented inBanicello et al.(2011) to include GLWB guarantees. They price GLWB guarantees under the passive and dynamic contract holder behaviours and evaluate the fair charge’s sensitivity to various assumptions and guarantee features. They also compare GMWB and GLWB guarantees’ fair charges.

A similar framework to the one presented inHolz et al.(2012) is used inKling et al.(2011) to price and hedge GLWB guarantees. First using the Black-Scholes framework and then making an extension to include stochastic volatility through the use of the Heston model, they find the fair guarantee charges and greeks of GLWB guarantees for various guarantee designs assuming a passive contract holder behaviour. They also analyze the hedge effectiveness for different dynamic hedging strategies and examine the effects on hedge effectiveness if the hedging model is different from the data-generating model.

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stand-point, which differs from the dynamic contract holder behaviour assumption most often used in the literature. They argue that a risk-averse contract holder may not only consider the level of future expected cash flows in making decisions, but also the negative effect of consump-tion fluctuaconsump-tions. They use the Money’s Worth Ratio and the Annuity Equivalent Wealth to compare various investment opportunities, including a GLWB and a GLWB with ratchets. The concept of utility is also used in Azimzadeh et al. (2014), which uses a method that consists in first determining the contract holder’s withdrawal strategy and then feeding it into the pricing problem. Using a Markov regime-switching process, they show that when consumption is assumed to be governed by hyperbolic absolute risk aversion utility, optimal behaviour leads to a cost similar to the one under the passive contract holder behaviour. The impact of policyholder behaviour is further analyzed inKling et al.(2014). They consider various lapse strategies and study the impact on pricing, hedging and hedge effectiveness. An optimal stochastic control framework allowing to price a GLWB guarantee under the Black-Scholes framework and under a regime-switching Markov process is developed inForsyth and Vetzal (2014). Various guarantee features and contract holder behaviour assumptions are considered.

Finally,Fung et al. (2014) is interested in the impact of systematic mortality risk on GLWB guarantees. They analyze the impact of mortality risk on the unhedged gains and losses distribution.

1.5 Motivation

As made clear by the literature review presented in section1.4, much of the attention devoted to GLWB and GMWB guarantees has been focused on pricing. Risk management, on the other hand, has drawn somewhat less attention. Nevertheless, risk management is very important for GLWB guarantees, as these guarantees contain significant systematic risks and represent a large proportion of the insurance industry’s variable annuity inforce business.

Companies offering variable annuity guarantees have two opportunities at risk management. First of all, they can mitigate some of the risks associated with variable annuity guarantees through an appropriate pricing and product design. Adequate pricing helps preventing short-falls in the value of future revenues versus future claims. In other words, it ensures that the contract holder pays the fair price for the guarantee provided by the company. Guarantee design helps reducing the potential risks faced by insurers. For example, guarantee features that can have an impact on risk include the size of the roll-up, the frequency of ratchets, the lifetime withdrawal rates, the fund offering and so on.

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Indeed, GLWB guarantees by nature contain risks that cannot be fully mitigated by product design. Thus, the main part of risk management for variable annuity guarantees is carried out after the guarantees are issued. Then, prices charged for the guarantees can seldom be changed, and if so, only to a certain extent. Moreover, product design is usually carved in stone by the contractual agreement that binds the company and the contract holder. Companies thus have to find alternative ways to manage risks related to GLWB guarantees.

Options for risk management after issue of the guarantee are rather scarce for insurance com-panies. Self-insurance is certainly one of the least advisable risk management options. Since GLWB guarantees contain mostly systematic risks, this strategy could prove very damaging to a company with a large exposure to these guarantees. Examples of the potential adverse im-pacts on companies have been seen during the last financial crisis on even the largest Canadian insurers (see e.g. Tedesco(2010)).

Reinsurance also is a risk management option, but reinsurance opportunities for variable annuity guarantees have been rather rare since the last financial crisis. Thus, companies are often left with capital market hedging as the most viable risk management alternative after guarantee issue.

Since GLWB guarantees are complex, long-term, life-contingent options, it is not possible to find instruments on the financial markets that completely offset the sensitivity of these guarantees to their various systematic risk factors. The assessment of the residual risk of hedged GLWB guarantees thus becomes a critical step in the valuation of these guarantees. In this assessment, modeling holds a crucial place. Indeed, modeling is of particular relevance for variable annuity guarantees, as these guarantees are not marked-to-market like plain vanilla options, but rather marked-to-model.

The goal of this thesis is twofold. First of all, an analysis of how the GLWB guarantee liability modeling affects the hedge efficiency of GLWB guarantees is carried out. This analysis is in line with the hedge efficiency analysis presented in Kling et al. (2011). However, whereas

Kling et al. (2011) focused on the stock market model, this thesis aims to expand the scope to consider other very important systematic risks of GLWB guarantees, namely, interest rate and longevity risks. Interest rate risk is a major risk for GLWB guarantees as it affects both the average return in the risk-neutral valuation of the guarantee liability and the discounting of the guarantee’s long-dated cash flows. Longevity risk also is a significant risk because claim payments in GLWB guarantees are life-contingent. In this thesis, stochastic volatility for the stock market risk is introduced through the use of the regime-switching lognormal model, whereas it was done using the Heston model in Kling et al.(2011). Moreover, several interest rate models, including one-factor and two-factor models, are considered. Finally, longevity risk is accounted for through the use of a stochastic mortality model.

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impacts on hedge efficiency of actual deviations in the mortality experience are assessed. Deviations in the mortality experience are obtained through the use of stochastic mortality in the outer loop modeling of the hedge efficiency calculation. Moreover, a method that allocates the residual risk of hedged GLWB guarantees between financial and longevity risks is proposed. The method is further extended to split longevity risk into two risk components. Finally, diversification between financial and longevity risks for hedged GLWB guarantees is investigated. Hedge efficiency in projections in which both the financial and longevity risk factors are modeled in a single stochastic projection are compared to projections in which longevity risk is taken into account using an actuarial margin for adverse deviations on the mortality improvement assumption.

This thesis is organized as follows. In chapter 2, the notation and formulas related to the GLWB guarantee model considered throughout this thesis are introduced. The valuation of the GLWB guarantee liability is also presented. Chapter3 extends the GLWB guarantee liability valuation to the valuation at future points in time and to the greeks calculation. The models considered for the stock market, the bond market, interest rates and longevity, both in the outer and inner loops, are then introduced in chapter4. Chapter5first presents an unhedged real-world risk assessment of a GLWB guarantee, which leads naturally to introducing the hedging strategy considered in this thesis and illustrating how hedging can help mitigating the risks associated with GLWB guarantees. In chapter6, the impact of the guarantee liability modeling on hedge efficiency is thoroughly examined with respect to the stock market, interest rate and longevity risk factors. Finally, longevity risk in hedged GLWB guarantees is further analyzed in chapter7.

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Chapter 2

GLWB guarantee valuation

2.1 Introduction

In chapter 1, GLWB guarantees were described qualitatively. In this chapter, the GLWB guarantee considered in this thesis is described in a more formal manner, taking into account the additional features included in the guarantee.

Then, through the use of generic notation for some of the risk factors of GLWB guarantees, the valuation of the GLWB guarantee liability is presented. The guarantee liability valuation consists in determining, given the contractual charge set by the company for the guarantee, the expected present value of claims and revenues from the guarantee. Valuation is thus closely related to pricing, as pricing consists in setting a charge that is such that the expected revenues equals the expected claims from the guarantee. It is the calculation that is mostly dealt with in the GLWB guarantee literature. Valuation of the GLWB guarantee liability is the necessary foundation on which to build the risk assessment of hedged GLWB guarantees.

2.2 Guarantee model

Let SP be the single premium made in a variable annuity contract with a GLWB guarantee issued at time t0 = 0. Assume that t0 falls on January 1st of a given year for the sake of

simplicity. The contract holder’s age at inception of the contract is given by x. Let

ΩT = {t0, t1, . . . , tW −1, tW, tW +1, . . . , t(ω−x)/∆t} (2.1)

be the set of all ordered times at which events can occur, where tW is the time at which the

contract holder transfers from the accumulation phase to the withdrawal phase, ω is the maxi-mum age of the mortality table and ti+1−ti= ∆t ∈ (0, 1], ∀i. Events may include withdrawals

from the account value, ratchets and roll-ups of the guaranteed withdrawal balance, death of the contract holder or rebalancing of a hedge portfolio. Assume further that tW is an integer

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The contract holder is assumed to invest in a combination of mutual funds whose total value is given by the stochastic process

F = {Fti, ti ∈ ΩT}.

Flexibility in the contract holder’s asset mix is allowed by assuming that the fund is composed of two distinct mutual funds: a stock market mutual fund and a bond market mutual fund, each of which aims to track a given stock and bond index.

Let

FS = {FS,ti, ti ∈ ΩT} and

FP = {FP,ti, ti ∈ ΩT}

be the stochastic processes for the mutual funds that track the stock market index and the bond market index, respectively. Moreover, let

S = {Sti, ti ∈ ΩT} and

P = {Pti, ti ∈ ΩT}

be the stochastic processes for the stock market index and the bond market index, respectively. Rebalancing of the mutual funds between the stock market mutual fund and the bond market mutual fund is assumed to be done yearly at a given fixed proportion ψ. The proportion in each mutual fund within a given year depends on the performance of the stock market index and the bond market index in that year. Thus, the mutual funds values, for ti ∈ ΩT, are given

by FS,ti =      FS,ti−1 Sti Sti−1 e−mA∆t_, _t i6∈ N ψFti, ti∈ N , (2.2) FP,ti =      FP,ti−1 Pti Pti−1 e−mA∆t_, _t i6∈ N (1 − ψ)Fti, ti∈ N , (2.3) and Fti =      FS,ti + FP,ti, ti 6∈ N FS,ti−1 Sti Sti−1 + FP,ti−1 Pti Pti−1 e−mA∆t_, _t i ∈ N , (2.4)

where Ft0 = SP, FS,t0 = ψ SP, FP,t0 = (1 − ψ)SP and mA is the mutual fund management fee, that is to say, the contractual fee that is meant to cover commissions, investment expenses, and all other company-related expenses that are not related to the GLWB guarantee. The rebalancing strategy described above aims to consider the fact that contract holders generally

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have a target stock market proportion depending on their investment profile, but that they do not rebalance their portfolio frequently during a given year.

Let also

A = {Ati, ti ∈ ΩT},

be the stochastic process of the contract holder’s account value, and G = {Gti, ti ∈ ΩT},

be the stochastic process of the guaranteed withdrawal balance. As described in chapter 1, the guaranteed withdrawal balance is used to determine the lifetime withdrawal amount and depends on guarantee features such as ratchets and roll-ups.

At inception of the contract, the account value and the guaranteed withdrawal balance are equal to the single premium such that

At0 = Gt0 = SP.

The dynamics of A and G depend on the GLWB guarantee phase. The dynamic in each of these phases are given in sections 2.2.1 and 2.2.2.

2.2.1 Accumulation phase

As described in chapter 1, the accumulation phase is the phase that precedes the withdrawal phase and in which no withdrawals are made by the contract holder. Thus, the account value dynamic is given by Ati = Ati−1 Fti Fti−1 e−gA∆t_, _t i∈ ΩT and t0 < ti ≤ tW, (2.5)

where gA is the GLWB guarantee fee. The guarantee fee is the additional charge that covers

the costs related with the GLWB guarantee. It is distinct from mA, the mutual fund fee,

which only covers fund-related and company-related expenses. The guarantee fee can be seen as the revenue counterpart to the claims the company has to pay when the contract holder’s account value is exhausted. It is the charge that is usually solved for in the pricing exercise such that the expected value of claims equals the expected value of revenues.

The evolution of the guaranteed withdrawal balance during the accumulation phase depends on the particular features of the GLWB guarantee. As discussed in chapter 1, popular GLWB features in the Canadian and US markets include ratchets and roll-ups of the guaranteed with-drawal balance. Assume that the GLWB guarantee includes automatic ratchets every m years and yearly 100p% roll-ups of the guaranteed withdrawal balance. Then, the guaranteed with-drawal balance during the accumulation phase, that is to say, for ti ∈ ΩT and t0< ti< tW, is

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given by Gti =        max(Gti−1(1 + p), Ati), ti/m ∈ N Gti−1(1 + p), ti∈ N and ti/m 6∈ N Gti−1, otherwise . (2.6)

In (2.6), ti/m ∈ N implies that ti is an integer multiple of m years. Then, since ti is both at

the end of a year and at a ratchet time, both the roll-up and the ratchet features may alter the guaranteed withdrawal balance. Moreover, ti ∈ N and ti/m 6∈ N implies that ti is at the

end of a year, but not at an integer multiple of m years from the contract inception date. In this case, only the roll-up feature is applied to the guaranteed withdrawal balance.

Assume further that at the time of transfer from the accumulation phase to the withdrawal phase, an additional ratchet of the guaranteed withdrawal balance occurs. Then, since it is also assumed that tW ∈ N, the guaranteed withdrawal balance at the time of transfer is given

by

GtW = max(GtW −1(1 + p), AtW). (2.7)

Let R = {Rti, ti ∈ ΩT} denote the stochastic process of the company’s guarantee-related revenues, that is to say, the future revenue from the guarantee fee. In the accumulation phase, the revenue process is given by

Rtj = Atj 1 − e

−gA∆t , _t

i ∈ ΩT and t0≤ ti< tW. (2.8)

2.2.2 Withdrawal phase

In the withdrawal phase, the contract holder withdraws the lifetime withdrawal amount yearly. The dynamics of the account value and of the guaranteed withdrawal balance are therefore changed to take withdrawals into account.

Let L = {Lti, ti ∈ ΩT} be the stochastic process of the annual lifetime withdrawal amount. At the time of transfer from the accumulation phase to the withdrawal phase, the lifetime withdrawal amount is given by

LtW = GtW × lx+tW−t0, (2.9) where lx is the lifetime withdrawal rate at age x.

The contract holder generally can choose the withdrawal frequency, which can go from annual to monthly. Let n be the withdrawal frequency elected by the contract holder, with n = 1 for annual withdrawals and n = 12 for monthly withdrawals.

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The account value dynamic for ti > tW and ti∈ ΩT is then given by Ati =        max Ati−1 Fti Fti−1 e−gA∆t₋ 1 nLti−1, 0 , nti∈ N Ati−1 Fti Fti−1 e−gA∆t_, otherwise . (2.10)

As shown in (2.10), the account value may never become negative despite withdrawals. More-over, since the lifetime withdrawal amount may change at time ti as a result of a ratchet, the

amount withdrawn at time ti is based on the lifetime withdrawal amount determined at time

ti−1.

The guaranteed withdrawal balance is decreased on a dollar-for-dollar basis when withdrawals are made. However, there still is a possibility for a ratchet of the guaranteed withdrawal balance every m years in the withdrawal phase. The guaranteed withdrawal balance dynamic for ti > tW and ti ∈ ΩT is given by

Gti =              max Gti−1− 1 nLti−1, Ati , ti/m ∈ N max Gti−1− 1 nLti−1, 0 , nti∈ N and ti/m 6∈ N Gti−1, otherwise . (2.11)

Thus, the guaranteed withdrawal balance is increased in the accumulation phase as a result of ratchets and roll-ups, and then decreased in the withdrawal phase as a result of withdrawals, except for potential withdrawal phase ratchets. Like the account value, the guaranteed with-drawal balance may never become negative.

Although the lifetime withdrawal amount may never be decreased, hence providing protection to the contract holder, it may be increased in the event of a ratchet. The withdrawal amount for ti > tW and ti ∈ ΩT is given by

Lti =

(max Lti−1, lx+ti−t0Gti , ti/m ∈ N Lti−1, otherwise

. (2.12)

Note that the lifetime withdrawal rate used in determining the ratcheted lifetime withdrawal amount is based on the contract holder’s attained aged at the time of ratchet.

As time goes by in the withdrawal phase, withdrawals keep putting downward pressure on the guaranteed withdrawal balance. Since the latter is used in the calculation of the ratch-eted lifetime withdrawal amount, its decrease reduces the likelihood of a ratchet late in the withdrawal phase.

The revenue process in the withdrawal phase is very similar to the one in the accumulation phase. It is given by

Rtj = Atj 1 − e

−gA∆t , _t

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Clearly, no revenues are collected once the account value is exhausted.

Let C = {Cti, ti ∈ ΩT} be the stochastic process of claim payments from the company’s perspective, that is to say, lifetime withdrawal payments to the contract holder when the account value is exhausted. The claim payments for ti ∈ ΩT are given by

Cti =              max 1 nLtk∗−1− Atk∗−1 Fti Fti−1 e−gA∆t_{, 0} , ti= tk∗ 1 nLti−1, nti ∈ N and ti > tk∗ 0, otherwise , (2.14) where tk∗= (inf{t_k∈ Ω_T : Atk = 0}, Aω−x = 0 ω − x, Aω−x > 0 (2.15) is the time at which the contract holder’s account value falls to zero if it does, or the time necessary to reach age ω if it does not.

As made clear by (2.14), two events may lead to claims to the company. Firstly, the payment that reduces the contract holder’s account value to zero has to be partially paid by the com-pany. Second, all lifetime withdrawal payments after the account value is exhausted have to be paid by the company.

The GLWB guarantee model presented above illustrates the optionality related with GLWB guarantees. Indeed, by issuing a GLWB guarantee, the company agrees to pay the maximum between zero and an unknown series of payments that is dependent on the account value performance, and thus on financial markets. GLWB guarantees are thus considered complex options.

2.3 Guarantee liability

Insurance companies make commitments to contract holders through GLWB guarantees. As such, they must maintain a liability to ensure that they can fulfill their obligations towards contract holders. A liability in the general sense consists in the actuarial present value of future claims minus the actuarial present value of future revenues.

Although the term liability is seldom used for plain vanilla options, the calculation of the GLWB guarantee liability is somewhat similar to the valuation of an option. For plain vanilla options, the option premium is paid as a lump sum, and so the liability associated with the option consists only in expected future claims. The liability is then the expected discounted payoffs of the option. GLWB guarantees, on the other hand, are not financed by a lump sum premium. They are rather financed through a charge in percentage of the account value, the GLWB guarantee fee. Thus, the GLWB guarantee liability consists of two components. The

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future claims component is the expected sum of the discounted lifetime withdrawal payments that the company has to make once the contract holder’s account value is exhausted. The future revenue component is the expected sum of the discounted revenues from the GLWB guarantee fee.

The dynamics of several important processes for the valuation of the GLWB guarantee, in-cluding A, G, L, R and C, were defined in section 2.2. In this section, these processes are used in the valuation of the GLWB guarantee liability.

2.3.1 Notation

In order to derive an expression for the guarantee liability, some generic notation regarding some of the risk factors related with GLWB guarantees, that is the stock market, bond market, longevity and interest rate risks, shall be introduced. Specific models are not imposed at this point and will rather be the topic of chapter 4.

Stock market and bond market

The notation regarding the stock market index and the bond market index was already intro-duced in section 2.2, where

S = {Sti, ti ∈ ΩT} and

P = {Pti, ti ∈ ΩT}

were defined as the stochastic processes representing the stock market index and the bond market index, respectively. The notation used in this section is consistent with the notation defined in section 2.2.

Mortality and longevity

In traditional actuarial valuations, mortality and longevity are often factored in through a constant or projected deterministic mortality table. However, longevity risk, that is to say, the uncertainty around future mortality, is a systematic risk that is significant for GLWB guarantees. Hence, the notation introduced in this section takes into account the possibility of stochastic mortality. Let

µ = {µx,ti, ti∈ ΩT}

be the stochastic process of mortality forces, where µx,ti is the force of mortality that applies between time ti and time ti+1 at age x.

Reflecting longevity risk through the introduction of stochastic mortality implies that the future values of µx,ti are not known with certainty. Longevity risk thus must be distinguished from the mere randomness in future death times. Indeed, in this thesis, the fairly standard

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actuarial assumption that the insurer’s portfolio is composed of a large population of similar risks is made. Hence, conditional on µ, any remaining mortality risk is diversified away. To sum up, assume that the random variable Z represents the present value of a fixed life-contingent stream of cash flows at a known interest rate. Then, using the assumption made above, the unconditional expectation of the present value of cash flows is given by

E[Z] = EµE[Z|µ] = EµZ|µ . (2.16)

As shown in (2.16), the expectation is conditional on the mortality path, but once the mortality path is known then there is no further randomness in mortality.

Traditional actuarial notation can easily be modified to take into account stochastic mortality. For example, lettjp

(µ)

x,ti be the tj-year probability of survival for an individual aged x at time ti given a mortality path µ such that

tjpx,ti = Eµ h tjp (µ) x,ti µ i .

In this thesis, the terms mortality, mortality improvement and longevity will be used almost interchangeably.

Interest rates

Interest rate risk also is a systematic risk that impacts the valuation of the GLWB guarantee liability. The notation introduced thus once again considers the randomness in future interest rates. Let r(s), t0 < s < tω−x

∆t be the short rate at time s. The short rate is such that, under the risk-neutral Q-measure,

ZCM(t) = EQh_e−R0tr(s)ds i

,

where ZCM_(t)_{is the t-year zero-coupon bond price observed on the market.} 2.3.2 Guarantee liability valuation

As discussed previously, the guarantee liability can be split into two components: a future claims component and a future revenue component. Let Vt0 be the guarantee liability, V

C

t0 be

the expected present value of future claims and VR

t0 be the expected present value of future revenues at time t0. Clearly, the relationship between these variables is given by

Vt0 = V

C

t0 − V

R

t0. (2.17)

Since the GLWB guarantee liability valuation is akin to the valuation of a complex option, the calculation of the guarantee liability is done under the risk-neutral Q-measure. However, because stochastic mortality is also included in the valuation, the usual single expectation is

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replaced by a double expectation. The expression for the claims component of the guarantee liability is given by V_tC₀ = Eµ   E Q    (ω−x)/∆t X j=0 tjp (µ) x e − tj R t0 r(s)ds Ctj µ       = Eµ   E Q    (ω−x)/∆t X j=k∗₊₁ tjp (µ) x e − tj R t0 r(s)ds₁ nLtj−11{ntj∈N}−tk∗p (µ) x e −tk∗R t0 r(s)ds Btk∗ µ      , (2.18) where Btk∗ = Atk∗−1 Ftk∗ Ftk∗−1 e−gA∆t₋ 1 nLtk∗−1 (2.19)

and tk∗ is the time at which the account value is exhausted as defined in (2.15).

The three systematic risk factors considered are easily identified in (2.18). First of all, claim cash flows are related to the market performance through the GLWB guarantee model pre-sented in section 2.2. Secondly, the survival probabilities that are applied to claim cash flows are random based on the future mortality path. Finally, future interest rates are also random, which affects both the average market return and the discount factor under the risk-neutral measure.

Moreover, as is shown in (2.18), there is no account for lapses in the claims component. In the present work, the guarantee is valued using the passive contract holder behaviour assumption, which implies that contract holders do not make any excess withdrawals and do not lapse the guarantee. Qualitative arguments for the passive withdrawal behaviour assumption are presented in Shah and Bertsimas (2008). Moreover, Azimzadeh et al. (2014) show that for a large family of utility functions, the consumption-optimal strategy is close to withdrawing at the contractual rate. In all cases, the modeling of lapses is in no way a trivial issue and is not the focus of this thesis.

The future revenue component of the guarantee liability at time t0 is given by

V_tR₀ = Eµ   E Q    (ω−x)/∆t X j=0 tjp (µ) x Rtj e − tj R t0 r(s)ds µ       = Eµ   E Q    (k∗−1) X j=0 tjp (µ) x Atj 1 − e −gA∆t e − tj R t0 r(s)ds µ      . (2.20)

This expression once again emphasizes that the company has no revenues once the account value is exhausted.

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2.3.3 Guarantee liability valuation through simulation

The valuation of GLWB guarantees has been shown to be akin to that of complex options. Many types of options can be valued using closed-form formulas. However, as was shown in this chapter, GLWB guarantees have characteristics that distinguish them from standard or even mildly exotic options. These characteristics include contractual features such as ratchets and roll-ups, but also flexibility in terms of fund choices, which can include bond funds and diversified funds in addition to the traditional stock market funds. Therefore, in order to model realistic product designs, it is often necessary to use Monte Carlo simulation to value GLWB guarantee liabilities. Monte Carlo simulation is used in most of the GLWB guarantee liability valuation literature (see e.g. Kling et al. (2011), Holz et al. (2012) and Shah and Bertsimas (2008)). For a brief overview of Monte Carlo simulation from a general option valuation perspective, see sectionA.2in the appendix.

In section2.2, several processes related with the dynamic of the GLWB guarantee were defined: S, P , µ, F , FS, FP A, G, L, R and C. For a given process denoted by X, let

X(k)= {X_t(k)_i , ti ∈ ΩT}

be the kth _{stochastic realization of the process under the risk-neutral Q-measure. Let also}

r(k)= {r(k)(s), t0 < s < ω − x}

denote the kth _{realization under the risk-neutral Q-measure of the short rate path. Finally, let}

V_tC,(k)₀ , V_tR,(k)₀ and V_t(k)₀ denote the realizations in risk-neutral scenario k of the decremented

present value of claims, the decremented present value of revenues and the decremented present value of claims minus revenues respectively.

The valuation of the GLWB guarantee liability at time t0 by Monte Carlo simulation using

NI risk-neutral scenarios is presented in Algorithm 2.1.

Algorithm 2.1. Valuation of the GLWB guarantee liability at time t0

1. Simulate, under the Q-measure, stochastic realizations of the following processes: 1.1. Stock market process: S(1)_{, . . . , S}(NI);

1.2. Bond market process: P(1)_{, . . . , P}(NI); 1.3. Short rate path: r(1)_{, . . . , r}(NI); 1.4. Mortality process: µ(1)_{, . . . , µ}(NI).

2. Using the realizations in step 1, compute realizations of the following processes: 2.1. Stock mutual fund value process (using (2.2)): FS(1), . . . , FS(NI);

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2.2. Bond mutual fund value process (using (2.3)): FP(1), . . . , FP(NI);

2.3. Total mutual fund value process (using (2.4)): F(1)_{, . . . , F}(NI); 2.4. Account value process (using (2.5) and (2.10)): A(1)_{, . . . , A}(NI);

2.5. Guaranteed withdrawal balance process (using (2.6), (2.7) and (2.11)): G(1)_{, . . . , G}(NI); 2.6. Lifetime withdrawal amount process (using (2.9) and (2.12)): L(1)_{, . . . , L}(NI).

3. Using the realizations in step 2, (2.17), (2.18) and (2.20), compute realizations of the following liability-related values:

3.1. Future claims at t0: VtC,(1)0 , . . . , V C,(NI) t0 ; 3.2. Future revenues at t0: VtR,(1)0 , . . . , V R,(NI) t0 ; 3.3. Guarantee liability at t0: Vt(1)0 , . . . , V (NI) t0 .

4. Compute the estimator for the guarantee liability at time t0:

Vt0 = 1 NI NI X j=1 V_t(j)₀ .

Algorithm 2.1thus consists in simulating several paths of the relevant risk factors and using the GLWB guarantee model and the expression for the guarantee liability to compute the liability in each of these paths.

The simulation process for a given risk-neutral scenario can be illustrated by an example. Assume that a contract holder elects a 5-year accumulation period and quarterly withdrawals, and that a single initial deposit of $100,000 is made in the contract. The contract specifies that 5% roll-ups are applied on the guaranteed withdrawal balance and that the withdrawal rate at age 70 is 5.5%. Then, for a given risk-neutral scenario, the projection may look like the one presented in Table 2.1.

As shown in Table 2.1, the guaranteed withdrawal balance is increased yearly as a result of roll-ups. Starting at time 5, the contract holder receives an annual lifetime payment of 7,020, computed as the product of the 5.5% withdrawal rate and of the guaranteed withdrawal balance at time 5. Withdrawal payments reduce the account value to zero at time 18, time at which the company stops receiving revenues from the guarantee fee. The first claim payment is lower than the following payments, since the company can use the remaining account value to pay a part of this first payment. Then, the claim payments correspond to a fourth of the lifetime withdrawal amount since quarterly withdrawals are made. The claim and revenue cash flows illustrated in Table 2.1would then be combined with the discount factors and survival probabilities to obtain one stochastic realization of the guarantee liability.