Master Thesis. Variable Annuities. by Tatevik Hakobyan. Supervisor: Prof. Dr. Michael Koller

Transcription

1 Master Thesis Variable Annuities by Tatevik Hakobyan Supervisor: Prof. Dr. Michael Koller Department of Mathematics Swiss Federal Institute of Technology (ETH) Zurich August 01, 2013

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3 Acknowledgments I would like to express my gratitude to my supervisor Prof. Dr. Michael Koller for the support, useful comments and remarks, and his engagement through the learning process of this master thesis. A very special thanks goes out to the office staff of the Department of Mathematics of the ETH Zurich. They provided me with direction and technical support, and I appreciate all the instances in which their assistance helped me along the way. Furthermore, my sincere thanks goes to all my friends both in Armenia and in Switzerland. Thanks for always being there for me, during my ups and downs, for not letting me forget who I am and where I come from, and for making my days brighter even thousands of kilometers away from home. Also, I am truly grateful to Luys Foundation, with all its team members and scholars, for the assistance and help they have always provided me with. I have met many amazing and exceptional people there. And for the one who has made the many hours spent working on my thesis seem worthwhile after all, Armen. Last but not least, I thank my family: my parents Karen and Nune, and my sister Shushan, for instilling in me confidence and motivation during all times. Without their encouragement and faith in me, I would not have been even close to there were I am now. Thank You! 2

4 Abstract In these latter days life insurance industry has grown significantly by offering an enormous variety of life insurance products. Some of these products are aimed on providing investment opportunities. Here is when variable annuities become useful. Variable annuities combine the advantages of traditional life insurance products with unit-linked products. They have provided the insurance industry with the products that cannot be found elsewhere. Yet, with all the help there are many challenges the industry faces. This thesis focuses on providing a detailed overview of such variable annuities, address the challenges and explain how to deal with them. Commonly used terminology in the VA market will also be introduced. Variable annuity products have progressively grown both in the US and Europe. Moreover, these products have considerable duration, thus there are risks which should be monitored and reported continuously over a possibly long period of time. The market growth with the risks it carries is also described. Then, an analysis of customer outcomes, valuation, risk management and hedging is carried out. The thesis concludes with the analysis of some data. 3

5 Contents Acknowledgments 2 Abstract 3 1 Introduction Motivation Outline Variable Annuities What is a VA VA contract structure Types of VA GMDB GMAB GMIB GMWB Valuation of VA General preliminaries Finite models Continuous time models Valuation framework Static approach Dynamic approach Mixed approach The least squares Monte Carlo method Hedging of VA Introduction Hedging Strategies Self-insurance Reinsuring

6 4.2.3 Static hedging Dynamic hedging Black-Scholes Model The European put option Black-Scholes Model for the GMDB Black-Scholes Model for the GMAB Factors which impact value to risk What is risk Types of VA risks Policyholder behavioural risks in VA Lapse Annuitisation Withdrawal Regular premium Asset allocation Conclusion Data Analysis Introduction GAM model Analysis Using R Conclusions 53 Bibliography 55 5

7 1 Introduction Nowadays life insurance is essential in most leading countries. It has developed significantly and many variation of its products have been expanded to answer consumers needs. One of this products is variable annuity, which is the main topic of this thesis. Life Insurance Life insurance is a contract between insurance policyholder (insured) and the insurer, where the insured is promised to receive a specific amount of money in case of death or other events, depending on the contract. In return the policyholder pays a premium. Life insurance is a long term contract. As the contract details depend on the health of policyholder, there are many different types of insurances one can choose. For details on types of insurances we refer to [4]. First insurance company was founded in the United States in 1732, nevertheless life insurance was only added to its product line in And from that time on the insurance companies are concerned with any factor that may affect normal longevity. And once the contract has begun, and premiums are regularly paid by the policyholder, the company is at a risk on a permanent contract which it cannot break. From the collation of a vast amount of data, an assessment can be made of the rate of mortality or the likelihood of death occurring at each age. 6

8 Numbers can be quoted, but which individuals will die at each age cannot be stated. Consequently, all who pay life insurance premiums to the common fund do so with the same willingness that the fund shall be used to compensate the estates of those contributors at whatever age in life they may die, within their respective contract period. This is the basic theory of life insurance [15]. 1.1 Motivation Nowadays everybody knows or is somehow connected to life insurance. The modern life insurance industry is huge and it offers enormous amount of products. Many of these products provide investors with perfect possibilities, such as substantial investment guarantees, tax privileges, etc. Variable annuities are the most over-hyped and oversold products. The amount of annuity payments by investors grew rapidly from the time they were introduced. The reason is that the amount paid out in the payment phase depends on the underlying assets purchased in the investment phase, and the actual payout is linked to the price of the underlying assets (usually other forms of stocks and securities). If the price of the assets rises, the annuity payment rises as well; conversely, if the assets fall in price, the annuity payment falls as well. Our interest in variable annuities is both theoretical and practical. We want to show the actual mathematical representation of the models which help in hedging the annuities. Furthermore, we will try to combine the mathematical formulas with the real data and show the results on plots. Also as this industry is relatively new, there is still much to understand in managing risks generated by these products. 1.2 Outline This thesis is organized in the following way: Chapter 2: Variable Annuities Chapter 2 describes what is a variable annuity, history and the development of VA. We will also look at different types of annuities, try to explain how they differ from one another and when they are used. There are also many different types of VA contracts which will be discussed. Chapter 3: Valuation of VA 7

9 In this chapter we will focus on the valuation of VA. We will give some preliminaries, general definitions which are used in the valuation approaches. Discuss different approaches, both discrete and continuous settings. Quantification of risks and tools used to quantify these risks. And finally explain one of the useful simulations, namely Monte Carlo method. Chapter 4: Hedging of VA As the guarantees are the main characteristic of variable annuities, it makes pricing and hedging more complex. We will try to deal with it in this chapter. We will explain what hedging of variable annuities mean and how it can be achieved. We will look at Black-Scholes Model and apply the theory to separate account products such as variable annuities. As follows we will derive valuation formulae using option pricing for the guaranteed minimum death benefit, and guaranteed minimum accumulation benefit contracts described in chapter 2. Chapter 5: Factors which impact value to risk The chapter will mainly discuss managing risks. Firstly, we will explain what risk is and which are its factors. Then, we will try to estimate what happens when policyholder behaviour differs from accepted and how it affects the risk. Furthermore, we ll discuss and understand the fact that certain types of risks cannot be hedged, regardless of the hedging strategy. Chapter 6: Data Analysis Given data from certain insurance company we try to analyse and show how the theory works in practice. Before that we will give some preliminaries on GLM models, which were used in writing the code on R with the provided data. 8

10 Annuities are sold, not bought. 2 Variable Annuities First variable annuity products were used in US, they appeared in 1950s. The contract was issued by the Teachers Insurance and Annuities Association College Retirement Equities Fund. In Europe and Asia, on the contrary, the so called unit-linked products were used, which differed from VA by not providing investment guarantees. For about 40 years following the first issue, the growth of the VA market was gradual. Moreover, there has been an exponential growth of the market in terms of market size and complexity of the VA products. Strictly saying, unit-linked product is a life insurance product the investment benefits of which are directly in proportion to the number of units in a unit trust purchased on the policyholder s behalf. Nevertheless, they do not provide any investment guarantees. Variable annuities took the advantages of unit-linked products and combined them with the advantages of life insurance products with their longterm investment guarantees, thus creating the modern VA, as we know it. The benefits to the policyholder depend on the performance of the investment option. Typically, the benefit for a defined event (e.g. death, maturity) is higher of the value of the policyholder investment and the guaranteed amount. This implies that the policyholder is protected against the insured event (e.g. death) and poor investment performance. Of course the insurance companies will charge a premium (fee) for issuing these guarantees.[9] In some countries, the success of variable annuities is also due to tax incentives. This was implemented by the government to support the development 9

11 of individual pension solutions and contain public expenditure. Typically, tax reductions are granted during accumulation, so that a variable annuity can be considered a tax-deferred investment.[1] Let us give a graphic of the growth of the VA market during last 10 years and then proceed to the formal representation of variable annuities and its types. 2.1 What is a VA A variable annuity is a contract between you and an insurance company, under which the insurer agrees to make periodic payments to you, beginning either immediately or at some future date, usually when you retire. The insured can decide how to pay for the variable annuity contract. It can be done either by a single purchase payment or by a series of payments. By buying an annuity contract you must choose from range of investment funds. The value of your investment as a variable annuity owner will then vary depending on the behaviour of your investment fund. Investment funds for variable annuities are characterised as mutual funds. A mutual fund is a type of professionally managed collective investment vehicle that pools money from many investors to purchase securities, where a security is a 10

12 trade-able asset of any kind 1 [17]. Mutual funds may invest in many kinds of securities, for example stocks, bonds, money market instruments or other assets. Nevertheless, even though variable annuities are invested mostly in mutual funds, they differ from mutual funds in several ways: variable annuities let you receive periodic payments for the rest of your life (or the life of your spouse or any other person you designate). This feature offers protection against the possibility that, after you retire, you will outlive your assets. variable annuities have a death benefit. If you die before the insurer has started making payments to you, your beneficiary is guaranteed to receive a specified amount typically at least the amount of your purchase payments. Your beneficiary will get a benefit from this feature if, at the time of your death, your account value is less than the guaranteed amount. variable annuities are tax-deferred. That means you pay no taxes on the income and investment gains from your annuity until you withdraw your money. You may also transfer your money from one investment option to another within a variable annuity without paying tax at the time of the transfer. When you take your money out of a variable annuity, however, you will be taxed on the earnings at ordinary income tax rates rather than lower capital gains rates. In general, the benefits of tax deferral will outweigh the costs of a variable annuity only if you hold it as a long-term investment to meet retirement and other long-range goals.[18] VA contract structure At a basic level, an annuity is an insurance contract that provides regular payments for a specific period of time, like 10, 20, 50,... years or for policyholder s lifetime. Usually the contract has both savings and insurance components. 1 The United States Securities Exchange Act of 1934 defines a security as: Any note, stock, treasury stock, bond, debenture, certificate of interest or participation in any profitsharing agreement or in any oil, gas, or other mineral royalty or lease, any collateral trust certificate, pre-organization certificate or subscription, transferable share, investment contract, voting-trust certificate, certificate of deposit, for a security, any put, call, straddle, option, or group or index of securities (including any interest therein or based on the value thereof), or any put, call, straddle, option, or privilege entered into on a national securities exchange relating to foreign currency, or in general, any instrument commonly known as a security ; or any certificate of interest or participation in, temporary or interim certificate for, receipt for, or warrant or right to subscribe to or purchase, any of the foregoing; but shall not include currency or any note, draft, bill of exchange, or banker s acceptance which has a maturity at the time of issuance of not exceeding nine months, exclusive of days of grace, or any renewal thereof the maturity of which is likewise limited. 11

13 With a variable immediate annuity, the annuitant purchases the policy with lump sum payment and the policyholder begins to receive payments immediately. An immediate annuity may either be fixed or variable; that is, payments may remain constant throughout the life of the annuity (or the annuitant s natural life) or they may change according to the performance of the investments made by the lump sum payment. Furthermore, the policyholder chooses the rate of return on the selected funds, e.g. 0%, 3.5%, etc.. With a variable deferred annuity, the annuitant does not begin to receive payments until some future date. A deferred annuity has two phases: a savings phase and an income phase. During the savings phase, the annuitant places money into the annuity, which invests it on behalf of the annuitant. In the income phase, the annuitant receives payments. It is important to note that a deferred annuity is not taxed until the income phase begins. It also pays a death benefit to the survivor(s) of the annuitant. Nearly all retirement plans are deferred annuities. The base VA contract provides the policyholder with a tax-favored retirement savings vehicle that allows for investment choice and optional protection from the impact of market downturns. A VA typically includes some combination of the following structure of charges.[9] Upfront sales charge: typically expressed as a percentage of the purchase premium. Surrender charge: typically expressed as a percentage of the account value at the time of the contract surrender. Mortality and expense charge: typically expressed as a percentage of assets under management and deducted from the fund value on a daily basis. Individual fund expense charges: also expressed as a percentage of assets under management. Administrative charges: typically a flat (US dollar) amount deducted from the account value monthly. 2.2 Types of VA Variable Annuity is an Investment Fund with additional Insurance Guarantees. These insurance guarantees are often referred to as GMxB, Guaranteed Minimum Benefits of type x. They can be grouped into two main broad classes: - GMDB (Death) - GMLB (Living) 12

14 The second class can be arranged into three subclasses: - GMAB (Accumulation) - GMIB (Income) - GMWB (Withdrawal) In 1990s, insurance companies introduced the concept of Guaranteed Minimum Income Benefits (GMIB). With this contract the policyholders could convert their account value after certain period of time to an annuity using the conversion rates, which had been specified when the contract was signed. In the same period they also began offering Guaranteed Minimum Death Benefits (GMDB) to help VA policyholders with estate planning concerns. In the beginning this rider provided death benefit equal to the greatest of the premiums paid or the current account value at the time of the policyholder s death. But over time the benefit changed to provide for an increasing minimum death benefit equal to maximum of the guaranteed amount and the current account value. Moreover, every few years the guaranteed amount was due to be changed to the greater of the previous guaranteed amount and also the current account value of that specific year. In 2000, concept of Guaranteed Minimum Accumulation Benefits (GMAB) was issued. The concept of this benefit is that the policyholder is guaranteed to get back at least the minimum of all the deposits made, by the time the rider matures. In 2003, the living benefit Guaranteed Minimum Withdrawal Benefits (GMWB) was developed. In this case, the policyholder was able to withdraw any sum up to the annual amount (the guaranteed withdrawal base amount). Furthermore, such a withdrawal could be done only till this base amount was not equal to zero. Apart from some upfront costs, premiums are entirely invested into the reference funds chosen by the policyholder. As policyholder is the one who decides where to invest, the asset combinations can be both dynamic or conservative. Even when the contract is already in-force some guarantees can be added or removed. The cost of the guarantees, as well as other expenses, are expressed as a given percentage of the policy account value [1]. Benefits in VA Now let us give some details on each of the guarantees. For more details, please refer to [1]. 13

15 We will consider a single premium variable annuity contract to be issued at time 0. We will also assume that all possible guarantees are selected at time 0 and are kept for the contract duration. Also by our assumption, no partial withdrawal occurs prior to retirement time. Let us denote single premium as P. T 0 will be the end of accumulation period; so if T = 0 the annuity is immediate and hence some guarantees become meaningless. A t will be policy account (fund) value at time t. This value depends on the evaluation of the reference fund in which the single premium is invested GMDB Guaranteed Minimum Death Benefits, is a product which offers a guarantee in case of early death, during the accumulation phase and possibly for some years after retirement. Some insurers provide GMDB also after retirement, up to some maximum age. The structure of this guarantee is as follows: in case the death of the policyholder occurs prior to the stated maturity, the insurer will pay the greater between the account value and a guaranteed amount. Thus, in case of death at time t T (i.e. during the accumulation period), the benefit is given by b D t = max{a t, G D t }, t T, (2.1) where G D t is some guaranteed amount. This amount can be both fixed or dependent on the account value. Let us give couple of examples where the guaranteed amount is fixed. 1. Amount of premiums paid, net of partial withdrawals (return of premiums). Under our assumptions, with no partial withdrawals and single premium: G D t = P. (2.2) 2. The roll-up of premiums at a specified guaranteed interest rate: G D t = P e δt, δ > 0, (2.3) where δ is the guaranteed immediate interest rate. If we need to model situations where the roll-up feature applies only till a given age of the policyholder or the rate is adjusted at regular intervals, then we could modify the guarantee (2.3) to include a time-dependent, deterministic guaranteed interest rate. For guaranteed amount depending on the account value the examples are as follows. 14

16 1. The highest account value recorded at some specified time prior to death (ratchet guarantee): G D t = max t i <t A t i. (2.4) Here, the profits of the reference fund are locked-in at the ratchet dates t i, i = 0, 1,..., n and 0 = t 0 < t 1 <... < t n < T. 2. The account value at some specified date, known as reset date (reset guarantee): G D t = A max t, (2.5) i t i <t where times t i, i = 1, 2,..., n, now define the rest dates. The difference between the ratchet (2.4) and the reset (2.5) guarantee is in the behaviour of the guaranteed minimum amount: in the (2.4) the minimum amount never decreases, whereas a reduction can occur in the (2.5), if the account value decreases between two reset dates. Moreover, specifications of the guaranteed amount can be considered. If the contract does not include GMDB, the death benefit is just given by A t ; moreover, it can be still given by (2.1) by setting G D t = 0. (2.6) Combinations of guarantees are also possible; for example G D t = max{p e δ t, max t i <t A t i } (2.7) is a combination of roll-up and ratchet features.[1] GMAB Guaranteed Minimum Accumulation Benefits provide protection of the policyholder s savings account during the accumulation phase (i.e., the working period of the policyholder). At some specific date, typically, the end of the accumulation period, the insured, if alive, is credited the greater between the policy account value and a guaranteed amount, which we will define as G A t. Thus, the benefit is: b A t = max{a T, G A T }. (2.8) Here also, as it was with the GMDB, the guaranteed amount can be given by 15

17 1. the amount of premiums paid, net of partial withdrawals; 2. the roll-up of premiums, net of partial withdrawals, at a specified guaranteed interest rate; 3. the highest account value recorded at some specified times (ratchet guarantee). We would like to note that the reset guarantee was separated from itemization on purpose. The thing is that in the GMAB a reset option simply allows to postpone the maturity date T, which differs from the one described in the GMDB model. Hereby, 4. the reset guarantee, which gives the opportunity to renew the GMAB when it reaches maturity GMIB Guaranteed Minimum Income Benefits provide protection of the policyholder s savings account after retirement, in particular in the face of high longevity. This guaranty may be arranged in two different different ways. 1. The amount of annuity will be the greater between the fund value and a specified sum, G I T. The rate of annuitisation will be defined according to market conditions predominating at the annuitisation date. The periodic payment of the life annuity is expressed as b I = η max{a T, G I T }. (2.9) The guaranteed amount can be specified the same way as the GMAB was. Then, G I T is defined similarly by (2.1)-(2.7). 2. The rate of annuity will be the greater between the stated rate α and η. The benefit then is given by b I = A T max{η, α}. (2.10) This guarantee is usually referred to as Guaranteed Annuity Option (GAO). It is used if we deal with money, not paying attention to subjective preferences regarding annuitisation vs bequest, as well as asymmetric information. 16

18 Here the guarantee is for the amount to be annuitised. Before that, the contract is like an investment product, bearing some guarantee. Whereas, after annuitisation, the policyholder has no access to the fund value. The policyholder must choose the guarantee several years before contract goes into force. Typically, the GMIB may be exercised after a waiting period of 5-10 years. The cost of the guarantee is subtracting from the fund value during the accumulation period. If prior the beginning of the contract the policyholder decides to give up on it, the insurer stops deducting the relevant fee. Normally, full annuitisation is required: nevertheless, partial annuitisation is also admitted in some arrangements. Many other arrangements like capital protection or even money-back are available, providing a death benefit, which consists of residual principal amount. By saying this we mean that the annuitised amount net of the annual payment has already been cashed. The annual amount can be fixed, participating or indexed to inflation or stock prices. Let us note here, that the participating scheme of the annual amount never decreases, whereas in the indexed annual amount there is a financial risk, it can fluctuate in time. Also, the payment frequency can vary depending on the policy conditions. Let us consider several cases,that are most common in practice: (Ia) a whole-life annuity; (Ib) an annuity-certain with maturity T independently of life contingencies; > T, in which payments occur (Ic) an annuity-certain with maturity T > T, followed by a deferred wholelife annuity if the insured is alive at T. And we assume flat annuity payments, as given in the equations above.[1] GMWB Guaranteed Minimum Withdrawal Benefits, allow the policyholder to withdraw a certain amount of assets at certain dates and to receive minimum payments. These withdrawals can be made even if the account value reduces to zero. The guarantee concerns the periodic payment and the duration of the income stream. This periodic payment is actually the percentage of the base amount, W t. We will denote this percentage as β. Let us note here, that as we have assumed all guarantees to be selected at time 0, this would coincide with the single premium P. The withdrawal amount will be b W t = β t W t (2.11) 17

19 where t is from the set of fixed withdrawal dates. The guaranteed payment can also increase in time, this can happen in the ratchet guarantee. But still, in some contracts a limit for the increase in annual payments is stated. Alternatively, the guaranteed payment can be exact, the maximum or the minimum amount that the policyholder is allowed to withdraw. In the latter case, any withdrawal above the guaranteed level reduces the base amount. The duration of the withdrawals can be fixed, for example for 20 or 30 years, or lifetime. In case of fixed withdrawals, if at maturity the account value is positive, it is paid back to the policyholder, or the contract stays in-force until the exhaustion of the policy account value. We assume that withdrawals can occur (Wa) up to a date T > T, independently of survival; (Wb) up to a date T > T, only if the insured is alive; (Wc) during the whole life of the insured. The cost of the guarantee is deducted from the account value during the withdrawal period. If the policyholder cancels the guarantee the fee stops being applied. During this period the policyholder keeps access to unit-linked fund. Moreover, if at time of policyholder s death the account value is still positive, it is paid to his estate. GMWB is quite similar to GMIB in the sense of investment life insurance contracts, apart from the possible range of guarantees. The GMIB is like a traditional life annuity, whereas, the GMWB is a real innovation of variable annuities in respect of traditional life insurance contracts. Even more, many features like the presence of death benefits in GMIB or a lifetime duration for the withdrawals in the GMWB reduce their differences.[1] 18

20 3 Valuation of VA In this chapter we will evaluate the cost of guarantees under general model assumptions. We will focus on encompassing different VA benefits into general frameworks. We will also give some preliminaries on mathematics of VA, which are essential. We will describe some general approaches which bring us to the fundamental knowledge of VAs and also of its techniques, which are also compulsory. The stochastic elements in these models give us more precise information, but at the same time the algorithms and parameters become more complex and difficult to determine. Let us also note, that the corresponding theory is still in the development stage. 3.1 General preliminaries Modern financial mathematics is essential for understanding the concept of VA and how their models act. Let us begin with some general definitions. First we will give some definitions for discrete model and then for continuous Finite models Let (Ω, A, P ) be a probability space, where Ω is the finite state space, A - the family of all events, and the probability P (ω) > 0 for all ω Ω. For more details on probability theory we refer to [13]. Now let us consider a share. If we want to know the development of 19

21 the price of a share we need to use pricing theory. Also, in this chapter we consider all models to be finite. Definition 1. An option contract that gives its holder the right (but not the obligation) to purchase a specified number of shares of the underlying stock at the given strike price, on or before the expiration date of the contract is called call option. A European call option allows the holder to exercise the option (i.e., to buy) only on the option expiration date. An American call option allows exercise at any time during the life of the option [19]. Let us fix a time T, when all trading is finished. This also means that the shares are traded at times 0, 1,..., T. We will denote by F t the σ-algebra of the events observable at time t. Value of the shares at time t will be S(t). Thus, in case of stochastic processes k <, which represent the prices of the shares at 1, 2,..., k is S = {S t, t = 0, 1,..., T } with components S 0, S 1,..., S k. Setting S j t as the price of the jth share at time t, we say that S j is (F t ) t -adapted. This means that we know the price of S j 1 t. Next, we = (1 + r) t to be able to make risk free. fix an interest rate r. And set St 0 investments. Here the risk free discount factor is defined by β t = 1 St 0 Definition 2. A trading strategy is a previsible process 1 Φ = {φ t, t = 1, 2,..., T } with components φ k t, for φ t F t 1. For our case, φ k t is the number of shares of type k, which we have during [t 1, t). Therefore, φ t is called the portfolio at time t 1. Now knowing the value of the portfolio at time t 1, we want to determine it at time t. Thus, the return in the interval [t 1, t) is φ t S t, and hence the total return is the sum of them all till t, i.e. G t (φ) = t φ τ S τ, (3.1) τ=1 where S t = S t S t 1 is notation for vector valued stochastic process. Definition 3. A trading strategy is self financing 2, if φ t S t = φ t+1 S t, for t = 1, 2,..., T 1. 1 A previsible process is one which only depends on information available up to the current time, but not on any information in the future. 2 A portfolio is self-financing if there is no exogenous infusion or withdrawal of money; the purchase of a new asset must be financed by the sale of an old one. 20

22 Definition 4. A trading strategy is admissible if is self financing and { φt S V t (φ) := t if t = 1, 2,..., T φ 1 S 0 if t = 0 (3.2) is non negative. The set of admissible trading strategies is denoted by Φ. We have discussed here trading strategies with limited possibilities. It is not allowed to withdraw or add money, neither it is allowed to become bankrupt. Thus, this option can be used in cases where the trading strategies are similar to the description given above. Now let us proceed to the arbitrage opportunities. Definition 5. In financial mathematics, arbitrage 3 opportunities are defined as P (V t 0) = 1 and P (V t 0) > 0, where V 0 = 0 and V t denotes the portfolio value at time t. Definition 6. A mapping π : χ [0, ), X π(x) is called price system if and only if the following conditions hold: π(x) = 0 X = 0, π is linear. Here χ is the set of all contingent claims, which, in turn, is a positive random variable X. Also, if V T (φ) = X, the random variable X is attainable. And π = V 0 (φ) is the price of an attainable contingent claim. A price system is consistent, if π(v T (φ)) = V 0 (φ) for all φ Φ. The set of all consistent price systems is denoted by Π, and P denotes the set P = {Q is a measure equivalent to P, s.t. β S is martingale 4 w.r.t Q}, where β is the discount factor from time t to 0. The measures µ P are called equivalent martingale measures. 3 The simultaneous buying and selling of a security at two different prices in two different markets, resulting in profits without risk. 4 In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict future winnings. In particular, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values at a current time. [20] 21

23 Theorem 1. The following statements are equivalent 1. There is no arbitrage opportunity, 2. P, 3. Π. Theorem 2. Suppose there exists a self financing strategy φ Φ s.t. V 0 (φ) = 0, V T (φ) 0, E[V T (φ)] > 0. Then there exists an arbitrage opportunity Continuous time models Now let us assume that P holds for the continuous model. This is the main difference between discrete and continuous models. Definition A trading strategy φ is a locally bounded, previsible process. 2. The value process corresponding to a trading strategy φ is defined by V : Π R, φ V (φ) = φ t S t = k i=0 φk t S k t. 3. The return process G is defined by G = Π R, φ G(φ) = τ 0 φds = τ 0 k i=0 φk ds k. 4. φ is self financing, if V t (φ) = V 0 (φ) + G t (φ). Let us use the following notation for the proceeding theorem, also from [10]. The discounted value of share i will be Zt i = β t Si. i Whereas, the discounted return is G (φ) = k i=1 φi dz i. Thus, V (φ) = βv (φ) = φ 0 + k i=1 φi Z i. Definition 8. A trading strategy is called admissible, if it has the following three properties: 1. V (φ) 0, 2. V (φ) = V (φ) 0 + G (φ), 3. V (φ) is a martingale w.r.t. Q. Theorem The price of a contingent claim X is given by π(x) = E Q [β T X]. 2. A contingent claim is attainable V = V 0 + HdZ for all H. 22

24 Theorem 4. If φ Φ, then V (φ) is a positive local martingale 5, and also a supermartingale 6, under each Q P. Definition 9. The market is called complete, if every integrable contingent claim is attainable. 3.2 Valuation framework Let us consider three different assumptions concerning the policyholder behaviour which lead to corresponding valuation approaches. These approaches are called static, dynamic and mixed. We will closely follow the notation given in [1]. From 3.1 we know that the market price of a security is given by its expected discounted cash flows, of course under reasonable economic assumptions. The incompleteness of insurance markets implies infinitely many riskadjusted probability measures. In this manner, we assume that the insurer has picked a specific probability for valuation. Thus, from now on all random variable and processes will be considered under this probability. Also discounting is performed at a risk-free rate. We will use the following notation. The balance of the money market account formalising the investment of cash at the immediate risk-free rate r, at time t will be M t = e t 0 rudu, t 0, (3.3) where r t is the immediate risk-free rate at time t. Also, the expectation is denoted by E[. ], accordingly E[. ] is the expectation conditional on the information at time t. S t will denote the unit value at time t of the reference fund backing the contract. Finally, τ stands for the time of the death of the policyholder. To conclude the notation let us recall from chapter 2, b D t to be the lump sum benefit paid in case of death at time t and thus denote by Bt L the cumulative benefits paid in case the of survival up to time t. 5 An adapted RCLL (right continuous with left limits) process M is said to be a local martingale if there exists an increasing sequence of stopping times {T n } s.t. P{T n = T } 1 as n, and the stopped process M(t T n ); 0 t T is a martingale for each n, in which case the sequence {T n } is said to reduce M. For more details we refer to[8]. 6 A supermartingale is an adapted RCLL process X = {X t ; 0 t T } s.t. X t is integrable and E(X t F s ) X s for 0 s < t T. A process X is said to be a martingale if both X and X are supermartingales. 23

25 3.2.1 Static approach For this approach it is essential that the contract never surrenders. Also one of the conditions is for the policyholder not to withdraw from the account during the accumulation or the payout period of the GMIB. And in case of the GMWB, he can withdraw only the amount mentioned in the contract. Now let us specify each of the cumulative benefits separately. GMDB Death benefit was given in the previous chapter. Let us recall that benefit paid in case of death at time t T was given by 2.1. Now let us define death benefits for the proceeding cases. For GMWB, in case death occurs at time t > T, we have b D t = max{a t, R t } for T < t T in 3.5 max{a t, 0} for T < t T in 3.6 max{a t, 0} for t > T for 3.7 (3.4) where R t denotes the value at time t of the remaining by contract guaranteed withdrawal amounts due between t and T. And for survival benefits GMAB and GMIB, b D t = 0, when t > T. GMWB Here, we have cumulative survival benefit, which is given by (Wa) Temporary withdrawals independent of survival: B L t = m b W T i 1 t Ti + max{a T, 0}1 t T, (3.5) i=1 where the withdrawal dates are (T )T 1 < T 2 <... < T m ( T ). The reason for ignorance of the continuations of the contract after T until fund exhaustion, is the second term, which represents the payment at date T from the remaining account value, if the insured is alive. (Wb) Temporary withdrawals in case of survival: in this case the cumulative survival benefit is given exactly the same way as in 3.5, i.e. B L t = m b W T i 1 t Ti + max{a T, 0}1 t T, (3.6) i=1 where the withdrawal dates are (T )T 1 < T 2 <... < T m ( T ). 24

26 (Wc) Whole life withdrawals: B L t = b W T i 1 t Ti (3.7) i=1 GMIB where (T )T 1 < T 2 <... are the dates for withdrawals. Let us follow the order given in (Ia) Whole-life annuity: Bt L = b I 1 t Ti, (3.8) i=1 with annuity payment dates (T )T 1 < T 2 <.... (Ib) Annuity certain: m Bt L = b I 1 t Ti, (3.9) with annuity payment dates (T )T 1 < T 2 <... < T m ( T ). (Ic) Annuity certain followed by a deferred whole-life annuity: i=1 Bt L = b I 1 t Ti. (3.10) i=1 with annuity payment dates (T )T 1 < T 2 <.... The annuity payment b I here is also from 2.9 or GMAB The cumulative survival benefit can be expressed by { 0 if t < T Bt L = b A T if t T = ba T 1 t T, (3.11) where the benefit paid at T is given by 2.8 and 1 E is the indicator of the event E, which takes value 1 if E is true and 0 otherwise. Now let us proceed to the valuation. To compensate the cost of all the guarantees there is a fee rate applied to the account value. This proportional 25

27 fee rate will be denoted by ϕ. Thus, in our case for static approach, the immediate growth of the contract value during the life of the policyholder is interpreted as follows: da t = { At ds t S t ϕa t dt db L t if A t > 0 0 otherwise, (3.12) and A 0 = P, let us recall that P was denoted to represent the single premium. Consequently, the return on the account value is that of the reference fund, adapted for the fees and survival benefit. Now let us denote the total accumulated benefits paid by the contract up to time t by B t. We get B t = (B L τ + b D τ )1 τ t + B L t 1 τ>t, t 0 (3.13) where τ is the time of the policyholder s death. Finally, the value of the VA policy at time t is given by the expected discounted cash-flows V t = E t [ t M t M u db u ], t 0, (3.14) where M is defined by 3.3. Moreover, as V t is the net of the fees, the contract is fairly priced if and only if V 0 V 0 (ϕ) = P. (3.15) Let us mention here that in case of very complex assumptions it won t be possible to obtain the necessary formulae. In this case a straightforward application of the Monte-Carlo method can be carried out in order to value the expectation in Dynamic approach In contrary to the static approach, in case of the dynamic approach the policyholder does not necessarily need to withdraw amounts as specified in the GMWB contract. Also, the policyholder is free to decide not to withdraw or even to surrender it at any time. It can be done even when the time does not coincide with the one specified in the contract, as well as during the accumulation period. Nevertheless, mostly in all cases of withdrawals there are penalties applied to the amount withdrawn. This basically means that in case of surrender the amount received would be less than the account value, even less than in case of partial withdrawal. And finally the guaranteed amount would also be reduced. Now, as soon as the stochastic process specifying the amount to be withdrawn at each date is given, the equivalent cumulative survival benefit is 26

28 obtained by modifying Clearly, only withdrawal strategies that have contract limit are accepted. We need to consider two cumulative survival benefits processes, one with net withdrawal amounts (NB), the other instead including gross amounts (GB). Let us denote the specific withdrawal strategy by θ, which will be added as a superscript to all relevant symbols introduced before. Then corresponding net and gross cumulative survival benefits will be (NB) L,θ t and (GB) L,θ t accordingly. Hence, the instantaneous process of the account value in the dynamic approach will be da θ t = { A θ ds t t S t ϕa θ t dt d(gb) L,θ t if A θ t > 0 0 otherwise, (3.16) and here the A θ t = P. In case of a surrender of the contract, let us denote the surrender time by λ θ, and if the it is not applied by the withdrawal strategy λ θ = τ. Given θ, the contract value at time t is expressed as where B θ t = ((NB) L,θ τ V θ t = E t [ t M t M u db θ u], t 0, (3.17) + b D,θ τ )1 τ min{t,λ θ } + (NB) L,θ t 1 t<min{τ,λ θ } + (NB) L,θ 1 λ θ λ θ <τ,λ t, t 0. θ Finally the contract value, assuming rational policyholder behaviour, is (3.18) V 0 = sup V0 θ, (3.19) θ where the supremum is for all admissible withdrawal strategies. Let us mention here, that even though we assume the behaviour is optimal from the policyholder s prospective, it also corresponds to the worst case scenario for the insurer. We will closely discuss policyholder behaviours in the proceeding sections Mixed approach This is the approach, when there is a GMWB guarantee, the policyholder can withdraw only at the time specified in the contract. Moreover, no partial withdrawals are allowed during the accumulation period or payout phase in case of the GMIB guarantee. However, the policyholder can decide to surrender at any time during the life of the contract. This is the main difference between static and mixed approaches. 27

29 Let us denote the surrender value, net of the penalty that the policyholder receives in case of surrender at time t > 0 by b S t. If surrender is not permitted during the payout phase, then b S t = 0 for some t. This can happen in case of GMIB guarantees. Here, as surrender value is the only difference and hence addition to the static approach, we just need to add it to the total cumulative benefit formulas defined in the While 3.12 still holds. Let us here denote λ as the surrender time, where λ = τ if the policyholder never surrenders. Thus, the total cumulative benefits Bt λ generated by the contract up to time t, will become equal to B λ t = (B L τ + b D τ )1 τ min{t,λ} + B L t 1 t<min{τ,λ} + (B L λ + b S λ)1 λ<τ,λ t, t 0. (3.20) As you see, only the second line of the equation was added, which is exactly the surrender value we needed. The corresponding value of the contract is V λ t = E t [ t M t M u db λ u], t 0. (3.21) And finally, at time 0 the contract value, which is basically net of insurance fees, is given by: V 0 = sup V0 λ, (3.22) λ where the supremum is taken with respect to all possible surrender times. Moreover, the fair fee rate is same as before Also, to evaluate it we need a numerical approach. One of such methods will be discussed in the next sections, namely the Monte Carlo method. Let us conclude this section, by comparing the results of these three methods. It turns out that the contract values at time 0 are in the following relationship: V static 0 V mixed 0 V dynamic 0. We see that from the companies point of view, the worst case scenario is when dynamic approach is used, as policyholder s behaviour is also taken into account for valuation. Whereas, in the mixed approach, the policyholder can only choose to surrender. And finally, the static approach defines a single withdrawal strategy, which at the same time is included in the mixed approach. 28

30 3.3 The least squares Monte Carlo method In the previous sections we have described different valuation approaches. We have chosen to describe one of the possible methods for the evaluation of the value of the contract (net of insurance fees) by means of a numerical approach. The algorithm is based on the Least Squares Monte Carlo method (LSMC). This method combines Monte Carlo simulation with the least squares regression within the dynamic programming principle. For more details on the least squares regression algorithm we refer to [3], whereas our focus will be on the description of the LSMC algorithm. For the LSMC we need discrete time dimension. Without loss of generality, we denote the time grid by T = {0, 1, 2,..., N}, for some integer N depending on the contract type. A set of state variables relevant for the valuation problem must be chosen. These variables represent financial risk factors, like interest rates, reference fund, policy account values and etc., which affect the contract under close analysis. We denote by X the vector of state variables. The set of basis functions will be denoted by e 1,..., e k for some k > 0, where k Z. Each basis function maps the state vector X into the set of real numbers. Step 0 (simulation) Simulate H paths of X over the time grid T. With reference to the hth simulation (h = 1,..., H), we use the following notation: -X h t vector of state variables at time t T, -M h t money market account at time t T, -τ h time of death, τ h T, -Ft h cash flow from the contract at time t T. Depending on the riders included in the contract, it may involve: a death benefit b D,h t when t = τ h ; a surrender benefit b S,h t if τ h > t; and a survival L,h payment b t = B L,h t B L,h t 1 if τ h > t. Step 1 (initialization) Set N = max h=1,...,h τ h. For all h = 1,..., H, let F h τ h = b D,h τ h and λ h = τ h. Step 2 (dynamic programming) For t = N 1, N 2,..., 1: i. Set H t = {1 h H : τ h > t} and, for h H t, C h t = λ h u=t+1 F u h Mt h. Mu h 29

31 ii. Regress the values (Ct h ) h Ht against (e(xt h )) h Ht, i.e. find γ t that solves arg min γ R k H h=1 (Ch t γe(x h t )) 2 (where e = (e 1,..., e k ) is a vector-valued function and) to obtain Cˆ t h = γ t e(xt h ), h H t. If b S,h t > C ˆ t h then set λ h = t and Ft h = bl,h t + b S,h t, otherwise set Ft h L,h = b t. Step 3 (contract values) Compute the value of the variable annuity at time 0: V 0 = 1 H λ h H h=1 t=1 Ft h. Mt h Note that the algorithm starts at each simulated time of death and then proceeds backward, so that each process does not have to be simulated over the entire time interval. For more details or alternative algorithms we refer to [2]. 30

32 4 Hedging of VA As we know from previous chapters there are many types of different risks. When companies take on such risks regulators and rating agencies start to pay closer attention to them. As a consequence, whenever the insurance companies get a chance to run an internal hedging programme, they do so. 4.1 Introduction Until the late 1990s the risks related to guarantees were not hedged. And even though they were not reliable, only few companies still deploy that activity. Major losses come from basic risks. One of these risks is a tracking error, which impacts on the balance sheet at once, and many hedgers run a deltahedging programme and assume volatility risk. On the contrary to the risk mentioned above, many insurers decided not to hedge their volatility risk, even though volatility was believed not to be expensive. Yet, is was considered to be cheap till 2007, thanks to abundant liquidity in the capital markets [9]. Companies are free to make financial decisions not to hedge their volatility risks. Yet, internal hedging should not be used as a method to ignore certain financial risks. Internal hedging has also some hidden risks. Unlike dynamic hedging, core hedging is about robustly hedging longterm cashflows. When using core hedging insurance companies can return 31