Analytical Calculation of Risk Measures for Variable Annuity Guaranteed Benefits

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1 Analytical Calclation of Risk Measres for Variable Annity Garanteed Benefits Rnhan Feng Department of Mathematical Sciences University of Wisconsin - Milwakee fengr@wm.ed Hans W. Volkmer Department of Mathematical Sciences University of Wisconsin - Milwakee volkmer@wm.ed Abstract With the increasing complexity of investment options in life insrance, more and more life insrers have adopted stochastic modeling methods for the assessment and management of insrance and financial risks. The most prevalent approach in market practice, Monte Carlo simlation, has been observed to be time consming and sometimes extremely costly. In this paper we propose alternative analytical methods for the calclation of risk measres for variable annity garanteed benefits on a stand-alone basis. The techniqes for analytical calclations are based on the stdy of geometric Brownian motion and its integral. Another novelty of the paper is to propose a qantitative model which assesses both market risk on the liability side and revene risk on the asset side in the same framework from the viewpoint of risk management. As we demonstrate by nmeros examples on qantile risk measre and conditional tail expectation, the methods and nmerical algorithms developed in this paper appear to be both accrate and comptationally efficient. Key Words. Variable annity garanteed benefit, Asian option, risk measres, vale at risk, conditional tail expectation, geometric Brownian motion, integral of geometric Brownian motion, Hartman-Watson density, modified Bessel fnctions. 1 Introdction In recent developments within both insrance and banking indstries, risk metrics sch as vale at risk and conditional tail expectation have been employed in nearly all major capital reqirements and risk measrement models. Hence the accracy and efficiency in the implementation of these risk metrics become increasingly important for stakeholders in the indstries. The crrent market practice of implementing capital reqirements incldes factor-based formlas, stress and scenario testing, and stochastic approach, etc. According to Farr et al. 11, the most prevalent method is the stochastic approach, of which varios forms have been adopted by the 1

2 majority 57% of insrance companies that responded to the Tillinghast Enterprise Risk Management srvey and that calclate economic capital. As shown in their report, the stochastic approach sed in the insrance indstry generally refers to the implementation of Monte Carlo simlations. Althogh the approach is often applied to different financial measres for varios prposes of risk assessment, we can nevertheless smmarize the main procedre as follows. A set of ftre scenarios on a certain time frame is generated to reflect a company or reglator s view on the variability of economic otcomes. Under each scenario the economic otcomes are assessed according to certain acconting conventions to prodce relevant economic measres. Combining all scenarios, the economic measres are then ranked to form an empirical distribtion. A capital reqirement is then determined by an estimation of a chosen risk metric, sch as vale at risk or conditional tail expectation, applied to the distribtion of economic measres. Despite its engineering flexibility, the simlation approach is often resorce demanding and time consming. The reslts may sometimes be nreliable de to the sampling variability even with the aid of variance redction techniqes. Besides the potential problems with inaccracy and inefficiency, the simlation approach may also be ndesirable from the cost and benefit perspective. According to Farr et al. 11, 76% of the respondents in the srvey whose companies have more than $1 billion of annal revene se a form of stochastic approach. In contrast, only 27% of the respondents whose companies have less than $1 billion se a form of stochastic approach.... These statistics sggest that a flly robst economic capital model sing stochastic approach may present prohibitive cost and implementation challenges to small and medim sized insrers. There are many papers in recent actarial literatre regarding the pricing and valations of variable annity garantees sch as Bacinello et al. 3, Ballotta and Haberman 4, Coleman et al. 6, Van Haastrecht et al. 14, Wang 24, etc. nder varios eqity price and interest rate models. However, there is scarce literatre on risk assessment and management of variable annity garantees. Simlation appears to be the only available stochastic modeling tool in practice for risk capital calclation and sensitivity test of investment-linked prodcts, de to the lack of development on alternative techniqes and methodologies. No previos work has been done on the analytical calclation of risk measres, despite the widely acknowledged isses with Monte Carlo simlation. As the research on analytical methods for risk management of variable annities is in the early stage, this work is by no means an attempt to address the complex isse of model efficiency all at once. Mch more work needs to be done for more general and practical applications. Like many other analytical approaches, the application of methods in this work hinges on the tractability of nderlying models. Hence, we shall investigate an example of calclating risk metrics for two types of variable annity garanteed benefits nder the geometric Brownian motion asset pricing model, which is often the base model for simlation approach as well. In Section 2, we first introdce the basics of variable annity garanteed benefits and formlate the gross and net liabilities of these prodct designs in a probabilistic setting from an insrer s 2

3 perspective. Two most widely sed risk measres in practice are discssed in Section 3. De to the technical advances on exponential fnctionals of geometric Brownian motions in the past two decades, it becomes possible for s to derive analytic soltions to these measres for two most basic types of garanteed benefits. More details on the development of these techniqes in both actarial and finance literatre can be fond in Section 3. For reasons to be seen, we provide two alternative sets of soltions to facilitate practical implementation. The pros and cons of both approaches will become self-evident in Section 4, in which we enter discssions with a few nmerical examples. 2 Variable Annity Garanteed Benefits Variable annity is a type of accmlation annity that offers participation in the profit sharing of investment fnds. Prchase payments net of fees and charges are deposited in varios sbacconts of policyholders choosing. Each sbaccont is invested in a particlar fnd with a distinct investment objective. To protect investors from downside risk of fnd participation, insrers also sell garantee riders sch as garanteed minimm matrity benefit GMMB, garanteed minimm death benefit GMDB, garanteed minimm accmlation benefit GMAB, etc. Under a garantee rider, an insrer receives the proceeds from fees and charges, and is responsible for covering financial losses to policyholders in adverse economic scenarios. The risk management of these benefits often hinges on the insrer s ability to accrately qantify and assess its liabilities. Hence effective and efficient comptational methods and models for risk management are crcial to the development and maintenance of variable annity prodcts. Readers are referred to Hardy 16 for detailed acconts of variable annity garantees. A variety of eqity retrn models have been sed in the actarial and financial literatre for variable annity garantee prodcts, the most poplar being the lognormal model or also known as the geometric Brownian motion model. There are many benefits from adopting this model in the crrent context. 1 One can se the celebrated Black-Scholes formla to explicitly price variable annity garanteed benefits as well as to se dynamic strategies to hedge these option-like benefits. 2 It is also the basis of frther extension to the regime-switching lognormal model with mltiple regimes which are also well accepted in the indstry for economic scenario generation. See Hardy The lognormal model is one of the recommended models in the American Academy of Actaries report and has been tested and sed to generate the pre-packaged economic scenarios. See AAA 1, Appendix 2. 4 The lognormal model in which we compte risk measres for risk management prpose will be the same as the typical model in which the volatility smile/srface is estimated for pricing. Hence practitioners can assess the impact of eqity volatility for both pricing and risk assessment in a consistent manner and hence redces model risks. For these reasons, We shall assme in this paper that the dynamics of nderlying asset price is driven by a geometric Brownian motion. However, readers shold be reminded that, as every model has its own limitation, the lognormal distribtion is known to have lighter tails than the probability distribtions of eqity prices from empirical data. A comparative stdy of the lognormal and 3

4 alternative time series models can be seen in the aforementioned references. An analysis of risk measres nder other models shold be condcted in ftre research. We now illstrate how the liability distribtion is determined by the actarial approach, which assmes no dynamic hedging on investment garantees. The se of stochastic simlation for modeling the garantee liability was extensively stdied in Bacinello et al. 3, Baer et al. 5 and Chapter 6 of Hardy 16. To facilitate the comparison with the stochastic models in Hardy 16, we follow their notation for cash flow variables with slight modifications. G, the garantee level, typically ranging from 75% to 1% of the prchase payments nder the GMMB rider. The garantee may also accre compond interest p to an advanced age. This is referred to as a roll-p option often associated with the GMDB rider. F t, the market vale of the separate accont at t. F is considered to be the initial prchase payment at the beginning of the rider. For simplicity, we assme that no additional prchase payments or withdrawal is allowed. S t, the market vale of the nderlying eqity fnd at t. For simplicity, we assme the accont is invested entirely in one fnd. The asset price process in this fnd is defined, on a probability space denoted by Ω, P, {F t } t, by where B is a standard Brownian motion. S t S e µt+σb t, t >, 2.1 m, the annalized rate at which fees and charges are dedcted from the separate accont. Except for prchase payments based withdrawal charges, all contract fees and expenses are typically calclated and accred on daily basis. Since no withdrawal is considered on policy anniversaries, it is reasonable for s to model all fees and charges as being taken continosly. The portion available for fnding the garantee cost is called margin offset or rider charge and is sally split by benefit. In this paper, we denote the annalized rate of charges allocated to the GMMB by m e and that of the charges allocated to the GMDB by m d. Note that in general m > m e + m d to allow for other expenses. T, the target vale date or called matrity date, typically a rider anniversary on which the insrer is liable for garantee payments. L, the net present vale of ftre liabilities, disconted at a constant risk-free force of interest of r per year. The rate r generally is selected to reflect the overall yield on bonds backing the general accont of the variable annity writer. At the end of each trading day, the accont vale is adjsted according to the performance of fnds in which it invests and dedcted by mortality and expenses M&E fees and rider charges. Hence, withot the effect of investment garantees the accont vale at time t is given by F t F S t S e mt, t T, 2.2 4

5 and the margin offset income at time t is given by M t m x F t, t T, where m x is replaced with m e for the GMMB or m d for the GMDB. In this paper, we investigate two basic designs of investment garantee - garanteed minimm matrity benefit and garanteed minimm death benefit in their plain vanilla forms. Neither annal ratchet featres nor policyholder behavior is considered in this model. 2.1 GMMB A variable annity writer s incomes and liabilities are generally dependent on the srvivorship of policyholders. We set ot the actarial notation for modeling mortality. τ x, the ftre-lifetime of a policyholder at age x; t p x, the probability that a policyholder at age x will srvive t periods; µ x+t, the force of mortality for a policyholder at age x + t; The GMMB rider provides a minimm garantee on the balance of policyholder s separate accont at the time of matrity. From an insrer s point of view, the present vale of the added cost of a GMMB rider, which will be called gross liability in this paper, is given by the disconted payoff e rt G F T + Iτ x > T, with x + maxx,, and IA 1 if the statement A is tre or IA otherwise. It is sally assmed that policy exits de to deaths are independent of fnd performance. In Hardy 16, Chapter 6, the exits are treated deterministically so that the only sorce of randomness is the eqity price process. The rationale behind this may be explained as follows. According to the strong law of large nmbers, for a pool of N contracts of the same size with policyholders of age x at time where N is large enogh, there are approximately t p x N contracts still in force at time t. We might as well average ot the remaining liabilities at time t over all original contracts and treat them as if every contract redces to the portion t p x at time t of its original size. Hence, it is arged that the cash flow at time t for t T is generated as follows { t p x M t, t < T ; C t T p x G F T +, t T. Therefore the present vale of net loss is given by L T e rs C s ds + e rt C T. 2.3 Even thogh t p x has its probabilistic interpretation, it declines deterministically over time. 5

6 However, it is qestionable whether the strong law of large nmbers is applicable when modeling variable annity contracts. In most contracts, there is no standardized prchase payment and the size of payment varies largely by contract ranging from $1, to $1 million. When risk measres are calclated with the cash flow 2.3, the assmption is implicitly made nder the strong law of large nmbers that all contracts are of eqal size, say for instance, $1,. A single contract with a prchase payment of $1 million then shold be treated as 1 nits. Hence the length of all 1 contracts depends on the ftre lifetime of a single policyholder, which clearly violates the independence assmption of the strong law of large nmber. Since we do not assme all contract nits to be mtally independent in this paper, the calclations of risk measres are done on an individal contract basis. However, as one shall see later, it does not mean that we have to compte risk measres repeatedly for each contract in practice. Risk measres are in fact proportional to the size of contract see Remark 3.2 and hence one can qickly determine the risk measres for each contract by mltiplying risk measres per nit by appropriate nits of contract size. This linearity has nothing to do with whether or not the contracts are independent. The randomness of insrer s liability to each contract arises from two independent sorces the eqity price process and the ftre-lifetime of contract. Therefore the present vale of insrer s liability net of margin offsets from the GMMB, which will be called net liability in this paper, is formlated as T τx L e rt G F T + Iτ x > T e rs M s ds. 2.4 This concept essentially corresponds to the term present vale of accmlated deficiencies in the AAA report for each individal stand-alone contract. We shall leave the technical details and their distinctions in another paper. The fndamental difference between 2.3 and 2.4 is that the mortality is modeled by a deterministic fnction of time in the former and by a random variable in the latter. One shold note that the net liability formlated in Hardy 16 denoted by L here is in fact the conditional expectation of net liability in this paper with respect to the natral filtration of asset price process. T τx EL F T E e rt G F T + Iτ x > T e rs M s ds F T T t T T p x e rt G F T + tp x µ x+t e rs M s ds dt tp x µ x+t e rs M s ds dt T T p x e rt G F T + T sp x e rs M s ds L. Under the circmstances where the strong law of large nmbers does apply to contract length, i.e. all contract lifetimes are independent and of similar size, the average across all policies of aggregate liabilities converges to the expectation of individal liability and then or formlation redces to that of Hardy 16. 6

7 2.2 GMDB The death benefit is often determined by the greater of total prchases payments with roll-p and the accont vale at death. We assme that the death benefit is payable immediately at the time of death and that the garantee increases at a roll-p rate δ available p to the matrity. Note that the roll-p rate is sally offered at a modest rate less than the risk-free rate which reflects insrer s own rate of retrn on bonds investment, i.e. δ r. From an insrer s point of view, the present vale of the added cost of a GMDB rider, or called gross liability, is given by e rτ x e δτ x G F τx + Iτ x T. Similarly, the present vale of the GMDB gross liability less margin offsets, or called net liability, is determined by T τx L e rτ x e δτ x G F τx + Iτ x T e rs M s ds. 2.5 For practical applications, we consider that death benefits are payable on discrete time basis. Introdce the crtate ftre lifetime κ n x in years ronded to the pper one n-th of a year. κ n x : 1 n nτ x, where x is the integer ceiling of x. De to the time lag often reslted from claims and investigation, we assme the investment accont is accmlated and rider charges are dedcted p ntil the end of the one n-th year of the policyholder s death and the death benefit is payable at the end of the one n-th year. Then the net liability is given by L n e rκn x e δκn x n T κ G F n κ + Iκ n x x T e rs M s ds. 2.6 x When n is taken to be large, L n can be sed as a close estimate of L. In this paper we only consider the risk measres of the GMMB and the GMDB separately. However, the work can be easily extended with the combination of both riders. 3 Analytical Soltions to Risk Measres The two most common risk measres sed by practitioners are qantile risk measre, also known as vale at risk, and conditional tail expectation risk measre. The qantile risk measre for L is defined for α α 1 as V α : inf{y : PL y α}. Alternatively, since L is modeled by a continos random variable in this paper, we shall compte the qantile risk measre by V α sch that PL > V α 1 α

8 The qantile risk measre V α is interpreted as the minimm capital reqired to ensre that there is sfficient fnd to cover ftre liability with the probability of at least α. The conditional tail expectation risk measre for L is also defined for α α 1 as CTE α : EL L > V α. It is the capital reqired to cover the average amont of liabilities when they exceed the qantile measre with the probability of at most 1 α. There is extensive literatre on the pros and cons for each of the two measres. Since the calclation of conditional tail expectation depends on that of qantile risk measre, we develop comptational algorithms of both risk measres for GMMB and GMDB. Readers shold be reminded that these risk metrics are typically sed as measres of extreme events. The two formlations of liabilities L and L generally reslt in different vales of the risk measres, since the randomness of the policyholder s ftre lifetime also contribtes to the occrrence of extreme events. We also remark that the net liabilities gross liabilities are expected to be negative zero in most cases as sch prodcts are designed to be profitable. However, from the perspective of risk management, risk capitals are determined to ensre that sfficient fnds are available to cover nexpected losses in adverse scenarios and hence we are only concerned with the levels of risk measres corresponding to positive liabilities in worst cases, which form the basis of calclation for risk capitals. Ths in most parts of the paper we consider α to be greater than the probability of non-positive liabilities, which shall be denoted by ξ e for GMMB and ξ d for GMDB. 3.1 GMMB The formlation of gross liabilities provides a qantitative model for assessing market risk from the liability side of investment garantees whereas the formlation of net liabilities offers the qantification of combined effects of market risk on both liability and asset sides. Hence we treat them separately in two sbsections Gross Liability We first determine the probability that no garantee payment will be made at matrity. ξ e 1 PG F T, τ x > T 1 T p x P e mt S T G 1 T p x ΦlnG/F ; µ mt, σ 2 T. S F Throghot the paper we denote by Φx; a, b the distribtion fnction of a normal random variable with mean a and variance b. Most compting software packages offer bilt-in fnctions for normal distribtion of this form. Risk measres are generally mch easier to compte when no cash flows from margin offsets are inclded i.e. M s for s T, as the insrer s liability reslts only from the benefit at matrity. Under this assmption, risk measres serve as direct tools to assess the gross liabilities of 8

9 garanteed benefits. Althogh most literatre on the sbject has focsed on arbitrage-free prices of garanteed benefits, it is debatable whether no-arbitrage theory is really applicable as garanteed benefits are not tradable derivatives and there is no sizable market for hedging mortality risk. We want to point ot that risk measres of ftre gross liabilities can be sed as alternative pricing principles for rider charges. A comprehensive accont of premim principles in traditional life insrance can be fond in Rolski et al. 23 and Goovaerts et al. 13. Proposition 3.1. The qantile risk measre V α with α > ξ e for the gross liability of GMMB is given by V α e rt G F exp{µ r mt + σ T z β }, 3.2 where z β is the 1β% percentile of a standard normal distribtion and β 1 α/ T p x. Proposition 3.2. The conditional tail expectation CTE α with α > ξ e for the gross liability of GMMB is given by CTE α e rt G T p x F 1 α eµ r mt +σ2 T/2 Φz β ; σ T, Note that 3.2 and 3.3 are eqivalent to 9.7 and 9.17 in Hardy 16 for which no contract exits de to mortality are considered, i.e. T p x Net Liability While most research papers in the crrent literatre focs on the liability side for pricing and valation, we think it is also important to assess the revene risk on the asset side since most fees and charges are based on eqity-linking accont vales. In the extreme cases where eqity prices are low for a prolonged period, not only is the garantee liability high at matrity, the revenes generated from rider charges are also persistently low, which exacerbates the severity of losses to the insrer. The distribtion of liability net of margin offsets provides a tool to analyze the impact of financial risk on both the liability side and the asset side. Althogh the payoffs of variable annity garantees bear no resemblance to an Asian option, it trns ot that the net liability can be viewed as a similar path-dependent derivative. As we can see from the formlation 2.4 and 2.5, the distribtion of net liability is completely determined by ftre lifetime random variable, the geometric Brownian motion representing fnd vales at the time of payment, and its integral representing the accmlation of rider charges. The work by Yor 21 on the pricing of Asian option has led to the stdy of the joint distribtion of the geometric Brownian motion and its integral, which forms the basis of or analysis on variable annity garanteed benefits. A parallel stdy on the integral of geometric Brownian motion is done by the Eropean school of actarial scientists led by Goovaerts and coathors with applications in modeling annities with random investment retrns. Two distinct approaches are sed to derive a 9

10 doble Laplace transform of the integral of geometric Brownian motion in De Schepper, Goovaerts, Delbaen 9 and De Schepper et al. 1. Interestingly, Yor 22 provides an analysis connecting the aforementioned and other related reslts developed in both actarial and finance literatre. Interested readers are referred to Carr and Schröder 8 for comprehensive acconts of pricing methods for Asian options. Many other applications of these techniqes can be seen in Geman and Yor 12. With analogy to Asian options, we provide two methods of compting risk measres of garanteed benefits. As one shall see in the proofs, a key qantity to the calclation of qantile risk measre is the following probability distribtion. P t, w : P e rt F t + F Note that for y, t T PL > y τ x > T P e rt G F T e rs M s ds < w. 3.4 F e rs M s ds > y P e rt F T T + e rs M s ds < e rt G y F F F Ths, we can determine the probability of non-positive liabilities by ξ e 1 T p x PL > τ x > T 1 T p x P T, e rt F P. T, e rt y. F Proposition 3.3. The qantile risk measre V α with α > ξ e for the net liability of GMMB is determined implicitly by 1 α T p x P T, BT with BT e rt G V α /F and 2 2π 2 P t, w π 3 σ 2 t exp σ 2 t ν2 σ 2 t 8 exp 2w2 4πy w 2ρ ν σ 2 sinh y sin t σ 2 t 1 + ρ 2 + 2ρ cosh y exp A1 + ρ2 + 2ρ cosh y 2w ρ 2 dρ dy, 3.5 where ν 2µ m r/σ 2, A 4m e /σ 2. Remark 3.1. Note that P T, BT increases with V α, as P t, w increases with w de to 3.4 and BT decreases with V α. One can se the Newton-Raphson method to determine V α, for which we need P T, BT 2 2π 2 V α π 3 σ 2 T exp σ 2 T ν2 σ 2 T 8 exp 2y2 4πy BT 2ρ ν σ 2 sinh y sin T σ 2 T BT ρ 2 2 exp A1 + ρ2 + 2ρ cosh y F 2BT ρ 2 dρ dy. 1

11 The comptation of conditional tail expectation hinges on the vale of the following expectation. { Zt, w : P e rt F t t + e rs M } s ds I F F {e rt F t + t F e rs Ms. ds<w} F Proposition 3.4. The conditional tail expectation CTE α with α > ξ e for the net liability of GMMB is given by where Z is given by Zt, w w CTE α e rt F G T p x ZT, BT, α 2 2π 2 π 3 σ 2 t exp σ 2 t ν2 σ 2 t 8 2ρ ν ρ 2 + 2ρ cosh y exp exp A1 + ρ2 + 2ρ cosh y 2w ρ 2 2y2 σ 2 sinh y sin t and E 1 z is the exponential integral defined by E 1 z z e t /t dt. 4πy σ 2 t + Aρ ν E 1 A1 + ρ 2 + 2ρ cosh y 2w ρ 2 A few words of cation seem to be necessary regarding the comptation of integrals sch as 3.5 and 3.7. As pointed ot in Carr and Schröder 8 to similar integrals in the context of Asian options, we observe that the integrals are mltiplied by a constant { } 2 2π 2 π 3 σ 2 t exp σ 2, 3.8 t which cold be very large when σ and t are small and conseqently cases overflow problems for compting in extreme cases. In those cases, the integrals have to be compted with very high accracy which cold be difficlt with limited compting resorces. However, it does not seem to case mch concern for the GMMB since variable annity contracts are generally long-term, i.e. T. Under the circmstances when very small t and σ are reqired, we do have an alternative approach, which is to nmerically invert the Laplace transforms of P and Z. This approach is analogos to the Laplace transform approach for pricing Asian option as well. Proposition 3.5. The qantile risk measre V α for the net liability of GMMB is implicitly determined by 1 α T p x P T, BT with α > ξ e, BT e rt G V α /F and P t, w can be obtained by nmerically inverting its Laplace transform P s, w : e st P t, w dt given by P s, w 4 σ 2 w w ρ 2 /A ρ ν 1 { exp 1 } ρ ρ2 I 2η d dρ, 3.9 where 2η 8s/σ 2 + ν 2, A 4m e /σ 2, and I 2η is the modified Bessel fnction of the first kind. Similarly, we can provide an alternative approach to the comptation of conditional tail expectation risk measre. 11 dρ dy 3.7

12 Proposition 3.6. The conditional tail expectation CTE α with α > ξ e for the net liability of GMMB is determined by 3.6 in which the fnction Zt, w can be obtained by inverting the Laplace transform Zs, w : e st Zt, w dt with Zs, w 4 σ 2 w w ρ 2 /A ρ ν+1 + Aρν 1 { exp 1 } ρ z 2 I 2η d dρ 3.1 where 2η 8s/σ 2 + ν 2, A 4m e /σ 2, and I 2η is the modified Bessel fnction of the first kind. Remark 3.2. For variable annity prodcts, the garanteed benefit G is always qoted as a percentage of prchase payment F. It is clear from the constant BT in Propositions 3.3 and 3.5 that for each fixed α, the qantile risk measre V α mst be a linear fnction of F when all else are held constant. Similarly, the conditional tail expectation CTE α is also a linear fnction of F according to Propositions 3.4 and 3.6. Therefore, this remark may serve as a theoretical jstification of the sal practice of calclating risk measres in proportion to prchase payments. 3.2 GMDB With the techniqes shown in the previos sbsection, we can also obtain the risk measres for both the gross and net liabilities of the GMDB Gross Liability Note that the probability that there is no garantee payment over the term of the GMDB, denoted by ξ d, is given by T ξ d 1 T tp x µ x+t Pe δt G > F t dt 1 tp x µ x+t ΦlnG/F ; µ m δt, σ 2 t dt. We are interested in risk measres nder which the gross liability is positive. Proposition 3.7. The qantile risk measre V α with α > ξ d for the gross liability of GMDB is determined implicitly by 1 α T tp x µ x+t Φ ln eδ rt G V α ; µ r mt, σ 2 t F Proposition 3.8. The conditional tail expectation CTE α with α > ξ d for the gross liability of GMDB is given by G T CTE α e δ rt tp x µ x+t Φ ln eδ rt G V α ; µ r mt, σ 2 t dt 1 α F F 1 α T tp x µ x+t e µ r mt+σ2 t/2 Φ dt. ln eδ rt G V α ; µ r m + σ 2 t, σ 2 t F dt 12

13 3.2.2 Net Liability Similar to the actarial practice of payment periods in life insrance, we can also develop risk measres for the net liability based on death benefit payable at the end of one nth of a year, denoted by L n in 2.6. The net liability nder consideration in this sbsection refers to L n. The formlas can be easily extended to the net liability based on death benefit payable continosly, denoted by L in 2.5. However, in general, the implementation of risk measres for L n reqires less comptational efforts and can serve as good estimates for risk measres of L. The expressions for P t, w in the case of the GMDB are the same as those for P t, w in the case of the GMMB except that m e is always replaced with m d. Note that the probability of non-positive liabilities is determined by nt ξ d PL n 1 P κ n x k P L n > κ n x k n n k1 nt 1 k 1/np x 1/n q x+k 1/n P e rk/n e δk/n G F k/n k1 nt 1 k1 nt 1 k1 k 1/np x 1/n q x+k 1/n P e rk/n F k/n + F k k 1/np x 1/n q x+k 1/n P n, eδ rk/n G. F k/n k/n e rs M s ds > e rs M s ds < eδ rk/n G F F Proposition 3.9. The qantile risk measre V α with α > ξ d for the net liability of GMDB is determined implicitly by 1 α nt k1 k 1/np x 1/n q x+k 1/n P k/n, Bk/n, where P t, w is given by 3.5 with A 4m d /σ 2 and Bk/n e δ rk/n G V α /F. Proposition 3.1. The conditional tail expectation CTE α with α > ξ d for the net liability of GMDB is given by CTE α 1 nt 1 α k 1/np x 1/n q x+k 1/n e δ rk/n GP k/n, Bk/n F Zk/n, Bk/n, k1 where P t, w and Zt, w are given by 3.5 and 3.7 respectively, both with A 4m d /σ 2 and Bk/n e δ rk/n G V α /F. Since the comptation of risk measres for GMDB reqires the evalation of P k/n, V α and Zk/n, V α at mltiple points, there may potentially be overflow isses with the expression 3.8 in the cases where t k/n and σ are both very small. When that happens, we recommend the inverse Laplace transform approach which we shall reiterate in the following remarks for clarification. 13

14 Remark 3.3. The qantile risk measre V α with α > ξ d for the net liability of GMDB can be alternatively determined by 1 α nt k1 k 1/np x 1/n q x+k 1/n P k/n, Bk/n where P t, w is obtained by nmerically inverting P s, w in 3.9 with A 4m d /σ 2, Bk/n e δ rk/n G V α /F. Remark 3.4. The conditional tail expectation CTE α with α > ξ d for the net liability of GMDB can be alternatively determined by CTE α 1 nt 1 α k 1/np x 1/n q x+k 1/n e δ rk/n GP k/n, Bk/n F Zk/n, Bk/n, k1 where P t, w and Zt, w are obtained by nmerically inverting P s, w and Zs, w in 3.9 and 3.1 respectively, both with A 4m d /σ 2, Bk/n e δ rk/n G V α /F. 4 Nmerical Example In this section, we provide examples in which analytical calclations of qantile risk measre and conditional tail expectation for both GMDB and GMMB are implemented. Consider a variable annity prodct with both GMDB and GMMB riders designed for policyholders of age 65 at policy isse. The term of the variable annity is 1 years, i.e. T The GMMB rider provides the policyholder with the greater of a garanteed benefit, typically a percentage of initial prchase payment, and the accmlated vale in the sbaccont at matrity. 2. The GMDB rider offers a death benefit eqal to the greater of a garanteed benefit earning interest at a roll-p rate and the accmlated vale in the sbaccont. Both the death benefit and accont vale are accmlated ntil the end of the year in which the policyholder deceases. The death benefit is payable at the year end. To model the ftre lifetime of policyholders, we se the life tables pblished in an actarial stdy by the U.S. Social Secrity Administration 7 in 25. The stdy compiles a comprehensive set of period and cohort life tables by sex and calendar year based on historic and projected mortality data in the areas covered by the US social secrity program. For illstrative prpose, we se an excerpt of the period life table for male and calendar year 21 with the additional calclation of srvival probabilities as shown in Table 1. See the stdy 7, page 68 for the complete life table. 14

15 x q x k kp Table 1: Predicted mortality rates of a male at the age of GMMB The recommended risk measre for the determination of additional asset reqirement for eqitylinking prodcts by the American Academy of Actaries 1 is CTE 9%. According to SFTF 19, the recommended risk measres for the calclation of reserve and solvency capital by the Canadian Institte of Actaries nder varios circmstances are between CTE 8% and CTE 95%. Hence, in this example we provide reslts on both qantile and conditional tail expectation risk measres at the 8, 9 and 95 percentile levels. We se the lognormal model 2.1 for asset prices of the investment fnd in which the policyholder invests. The valation basis for this nmerical example is given as follows. Mean and standard deviation of log-retrns per annm are µ.9 and σ.3 respectively; Risk-free discont rate per annm r.4 ; M&E fees and rider charges per annm m.1; GMMB rider charge is 35 basis points per annm of the separate accont, i.e. m e.35. The range of annal rider charges for varios types of garanteed benefits in the North American market can be seen in Sn 2. Since modeling assmptions of parameters are proprietary information and not available to s, the set of parameters we chose is arbitrary bt within a reasonable range. Practitioners are encoraged to test the methods and algorithms proposed in this paper with their company-specific model assmptions. The calclation of risk measres for an individal contract is carried ot for varios combinations of garanteed benefit and initial prchase payment in Table 2. The garanteed death benefit and risk measres are all represented as percentages of initial accont vale. A simple jstification for 15

16 G/F 75% 1% 12% V 8% /F % % % CTE 8% /F 6.911% % % V 9% /F % 12.55% % CTE 9% /F % 3.296% 43.72% V 95% /F % % % CTE 95% /F % 4.41% % Table 2: Comptation of risk measres for the GMMB rider this practice can be seen in Remark 3.2. In this example, we sed the direct calclation method introdced in Proposition 3.3 to determine qantile risk measres. In particlar, the Newton- Raphson method is the root search procedre sed in or program for the exact reslts of qantiles. All figres are given in Table 2 with five decimal places with no ronding. We sed ten digits in actal comptations and qantile risk measres with all ten digits are sed in the calclation of conditional tail expectation based on Proposition 3.4. The cases marked with asterisks correspond to negative risk measres of net liability L, the calclation of which reqires an extension of analysis from previos sections. However, since risk capitals are always held in non-negative amont in practice, exact amont of negative risk measre is of no relevance for real-life applications. In these cases, we instead compte the qantile risk measre and conditional tail expectation of L : max{l, }. In the case where L has a probability mass at zero and α < ξ e PL, we shall define the conditional tail expectation of L as follows. CTE α L : EL IL > 1 α 1 ξ eel L > 1 α 1 ξ ecte ξe L. 1 α For example, in the case where G/F 75%, we observe that P 1, , which implies ξ e 1 1 p 65 P 1, %. Ths, the L has a probability mass at and the 8% conditional tail expectation of L is compted by CTE 8% L CTE %L %. The compting time may vary greatly according to compting software, compter configration as well as the choices of initial vale for root search procedres. All comptations in this example are done with the Maple 14 software package on a personal compter with Intel Corel 2 Do CPU 3.GHz and 3.25 GB of RAM. It takes between one and three mintes to prodce each reslt in Table 2 with this compter set-p Accracy and Efficiency Test We provide an example to compare the accracy and efficiency of three different methods, which are 1 direct calclation, proposed in Propositions 3.3 and 3.4; 2 nmerical inversion of Laplace 16

17 transform, proposed in Propositions 3.5 and 3.6; 3 Monte Carlo simlations, which is the eqivalent of crrent indstrial practice, tailored to individal contract valation. All methods are tested nder the same valation basis as given in previos examples with G/F 1%. For the second method, the Laplace transform P s, y is calclated according to Proposition 3.5 and the expression B.1. We then invert the Laplace transform nmerically sing the Gaver- Stehfest method. In simple words, the Gaver-Stehfest method approximates the vale of P t, y by where P t, y : n wk, n P k t, y, k1 wk, n : 1 n k k!n k!, Pk t, y : ln 2 2k! t k!k 1! k n k k 1 j P j j k + j ln 2 According to Abate and Whitt 2, the nmber of digits precision reqired for programming with the Gaver-Stehfest method increases with the nmber n. For small n we need roghly 2n digits precision for calclation. t, y. In or compting rotines sing Maple, we often invoke the NAG nmerical integration procedres, d1amc and d1akc, which only allow precision p to 15 digits. Therefore, we only se n 7 in the following example. A shortfall of this method is that there is no known error analysis according to Abate and Whitt 2. Nevertheless, we can also perform the first method of direct calclation for verification. Becase the two methods are sfficiently different, we can be certain of its accracy if the reslts from both methods agree p to a sfficient nmber of digits. The comptation of CTE is carried ot similarly by inverting the Laplace transform Z given by Proposition 3.6 with the Gaver-Stehfest method. For the third method, we estimate the risk measres throgh repeated sampling in two steps. First, 1, observations of the ftre lifetime random variable are generated for each basic experiment. For each of those vales that srpass the term of the GMMB rider, we simlate the stock prices for 25 trading dates each year for the whole term1 years and then determine the net liabilities according to 2.4. Since the negative vales of net liabilities do not affect the risk measres, we do not generate stock prices for those ftre lifetimes shorter than matrity in order to improve simlation efficiency. Then we se the 9, -th order statistics and the sample mean of the 9, -th throgh 1, -th order statistics as the estimators for V 9% and CTE 9% respectively and both estimators are applied to the same data set of net liabilities for each basic experiment. Second, the basic experiments are repeated 5 times. The third colmn of Table 3 reports the means of 5 observations of both risk measres. Since the tre vales of risk measres are confirmed by the first two colmns, we can evalate the mean sqared errormse of the estimators. The MSEs of the qantile and CTE estimators are and respectively. Since both estimators are prodced simltaneosly, we only state the rnning time once nder the qantile estimator. Table 3 reports on final reslts from all three methods as well as comptational times in mintes which indicate their relative time consmption. The Newton-Raphson method is sed in search for 17

18 Methods V 9% /F 12.55% 12.55% % Initial vale 1% 12%, 14% - Time mins CTE 9% /F 3.296% 3.296% 3.463% Time mins Table 3: A comparison of three comptational methods qantile in direct calclation and bisection method in inversion of Laplace transforms. It is clear from the comparison that the analytical methods are by far more efficient and accrate than Monte Carlo simlations Sensitivity Test In what follows, we perform sensitivity tests on each of the parameters involved in the comptation of risk measres while all else being the same as listed in the valation basis. The garanteed benefit level is set to be 1% of the initial prchase payment, i.e. G F. Only the Newton- Raphson method is employed in the following comptations. The middle colmn corresponds to the base case sed for the accracy and efficiency test. µ V 9% /F % 12.55% % CTE 9% /F % 3.296% % σ V 9% /F.4771% 12.55% % CTE 9% /F % 3.296% 43.37% m V 9% /F % 12.55% % CTE 9% /F % 3.296% 32.23% m e V 9% /F % 12.55% % CTE 9% /F % 3.296% % r V 9% /F % 12.55% 1.8% CTE 9% /F 37.37% 3.296% 24.57% Table 4: Sensitivity analysis of risk measres for the GMMB rider The reslts on sensitivity analysis in Table 4 can be interpreted as follows. The insrer s 18

19 liability rises V, CTE de to higher costs of garanteed benefits when the expected vale of log-retrns in the separate accont decreases µ, or the eqity prices become more volatile σ. Higher roll-p rates δ give rise to more generos benefits and hence indce higher levels of risk V, CTE. Althogh it seems conter-intitive that higher M&E charges m lead to higher risk measre V, CTE, one shold be reminded that only the margin offset allocated to matrity benefit m e is inclded in the calclation of margin offset income M s. While the margin offset m e is held constant, higher M&E charges do not cont towards the incoming cash flow for the GMMB rider bt rather redce the fnd vale in the separate accont and hence lower the incoming cash flow. Conseqently higher garanteed minimm payments are more likely to be paid ot, which raise both risk measres. The higher margin offset m e increases the incoming cash flow and hence leads to less liability and smaller risk measres V, CTE. We also change the vales of risk-free discont rate δ to illstrate the impact of disconting factor. As one wold expect, the present vales of liability become smaller V, CTE as the discont rate increases r. The risk measres appear to be very sensitive to the volatility coefficient σ. In the case where σ.2, we set the digits of precision in Maple to be 2 and the comptation takes abot five mintes for each of the two risk measres. For the rest of cases we se the 1 digits of precision by defalt and the rnning times vary between one and three mintes. The 9% CTEs in this example appear to be higher than what are often observed for C3 risk-based capitals in practice. However, this does not necessarily sggest that the risk capitals are generally nderestimated in the variable annity indstry. It shold be noted that risk measres are typically calclated at the aggregate level in practice as opposed to the individal level considered in this paper. The diversification of risks among contracts may redce risk measres significantly. One shold also note from the nmerical test that the risk measres are very sensitive to the volatility coefficient σ. A choice of smaller σ wold lead to mch smaller CTE amonts. 4.2 GMDB We now present the evalation of risk measres for the GMDB rider. One shold note that the comptational efforts reqired by the GMDB rider are significantly increased as P k/n, V α in?? is compted mltiple times and for very small k/n. Note that the constant 3.8 with t 1 and σ.3 is on the order of 1 96 and hence the doble integrals in Propositions 3.9 and 3.1 need to be compted accrately p to at least the order of 1 11 in order to maintain precision p to five decimal places in the final reslts. Therefore, we shall se nmerical inversion of Laplace transforms otlined in Remarks 3.3 and 3.4, which demands mch less comptational efforts in this case. The valation basis for the base nmerical example is given as follows. Mean and standard deviation of log-retrns per annm are µ.9 and σ.3 respectively; Risk-free discont rate per annm r.7 ; 19

20 Roll-p rate per annm δ.6; M&E fees and rider charges per annm m.1; GMDB rider charge is 35 basis points per annm of the separate accont, i.e. m d.35; Srvival model follows Table 1. G/F 75% 1% 12% V 8% /F % % % CTE 8% /F 7.185% % % V 9% /F % % % CTE 9% /F 14.37% 33.76% % V 95% /F % % 5.732% CTE 95% /F % 5.39% 69.14% Table 5: Comptation of risk measres for the GMDB rider Table 5 reports on the risk measres for varios initial garantee levels. We sed the bisection method to search for the qantile with accracy p to five decimal places. In a manner similar to the previos sbsection, the cases with asterisks correspond to conditional tail expectations of L n : max{l n, }. For example, the probability of non-positive net liability for the case of G/F 12% is given by ξ d Hence, CTE α L n 1.869CTE 86.9%L n %. The large impact of initial garantee on risk measres is de to the fact that the roll-p rate is set at a relatively high level and hence the effect of higher initial garantee are more prononced as the garantee amonts at later policy anniversaries accmlate rapidly. The big difference between qantile and CTE risk measres seems to sggest that the distribtion of net liabilities may have a heavy tail. Since the 9% CTE amonts are sed for determining capital reqirements in practice, we perform sensitivity tests on varios parameters for the 9% level as shown in Table 6. We change one parameter at a time while holding all other parameters constant as in the base case. The initial garantee is set at G/F 1. The middle colmn corresponds to the base case. As one wold expect, the pattern of risk measres with changes in parameters for the GMDB rider is very similar to that for the GMMB rider. Here we point ot the impact of the roll-p rates. The risk measres increase V, CTE with roll-p rates, since higher roll-p δ leads to higher garanteed benefit on all policy anniversaries and hence higher present vale of liabilities. The Maple codes for all nmerical examples are available pon reqest. 2