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This article was downloaded by: [University of Liverpool] On: 09 October 2014, At: 11:37 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK North American Actuarial Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/uaaj20 Valuation of a Guaranteed Minimum Income Benefit Claymore Marshall AIAA a , Mary Hardy FSA, FIA, CERA, PhD a & David Saunders PhD a a Department of Statistics and Actuarial Science , University of Waterloo, 200 University Avenue West , Waterloo , ON N2L 3G1 , Canada b Financial Risk Management in the Department of Statistics and Actuarial Science , University of Waterloo, 200 University Avenue West , Waterloo , ON N2L 3G1 , Canada c Department of Statistics and Actuarial Science , University of Waterloo, 200 University Avenue West , Waterloo , ON N2L 3G1 , Canada Published online: 27 Dec 2012. To cite this article: Claymore Marshall AIAA , Mary Hardy FSA, FIA, CERA, PhD & David Saunders PhD (2010) Valuation of a Guaranteed Minimum Income Benefit, North American Actuarial Journal, 14:1, 38-58, DOI: 10.1080/10920277.2010.10597576 To link to this article: http://dx.doi.org/10.1080/10920277.2010.10597576 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

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Page 1: Valuation of a Guaranteed Minimum Income Benefit

This article was downloaded by: [University of Liverpool]On: 09 October 2014, At: 11:37Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

North American Actuarial JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/uaaj20

Valuation of a Guaranteed Minimum Income BenefitClaymore Marshall AIAA a , Mary Hardy FSA, FIA, CERA, PhD a & David Saunders PhD aa Department of Statistics and Actuarial Science , University of Waterloo, 200 UniversityAvenue West , Waterloo , ON N2L 3G1 , Canadab Financial Risk Management in the Department of Statistics and Actuarial Science ,University of Waterloo, 200 University Avenue West , Waterloo , ON N2L 3G1 , Canadac Department of Statistics and Actuarial Science , University of Waterloo, 200 UniversityAvenue West , Waterloo , ON N2L 3G1 , CanadaPublished online: 27 Dec 2012.

To cite this article: Claymore Marshall AIAA , Mary Hardy FSA, FIA, CERA, PhD & David Saunders PhD (2010)Valuation of a Guaranteed Minimum Income Benefit, North American Actuarial Journal, 14:1, 38-58, DOI:10.1080/10920277.2010.10597576

To link to this article: http://dx.doi.org/10.1080/10920277.2010.10597576

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shall not beliable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out ofthe use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Valuation of a Guaranteed Minimum Income Benefit

38

VALUATION OF A GUARANTEED MINIMUM

INCOME BENEFITClaymore Marshall,* Mary Hardy,† and David Saunders‡

ABSTRACT

With a deferred variable annuity the policyholder pays an upfront premium to the insurancecompany, which is then invested in the financial markets for many years (the accumulation phase)until the policyholder decides to convert their investment (often at retirement age) into a streamof variable annuity payments. A Guaranteed Minimum Income Benefit (GMIB) is an option thatmay be included at inception of a variable annuity contract that, in exchange for small feescharged by the insurer, gives the policyholder a right to receive a guaranteed minimum level ofannuity payments upon annuitization. A GMIB is an attractive option because it protects thepolicyholder’s investment against poor market performance during the accumulation phase.

The value of a GMIB is affected by investment account returns, interest rates, and mortality.The intention of this paper is to value a GMIB in a complete market, focusing on the sensitivityof the GMIB value to the financial variables. Mortality is not incorporated into the valuation. Wepresent a comprehensive sensitivity analysis of the model employed. We decompose a GMIBpayoff, which is rather complicated, to analyze what drives the value of a GMIB. Our approachoffers a simple but effective way for insurers to measure the value of the GMIBs they offer, and itprovides insights into the risk management of GMIBs and other guarantees that provide similarpayoffs. Our model suggests that the fee rates charged by insurance companies for the GMIBoption may be too low.

1. INTRODUCTION

With a deferred variable annuity the policyholder pays an upfront premium to the insurance company,which is then invested in the financial markets for many years (the accumulation phase) until thepolicyholder decides to convert their investment (often at retirement age) into a stream of variableannuity payments. The magnitude of the annuity payments depends on the accumulated value of thepolicyholder’s investment at the date of annuitization. In the U.S. annuity market, embedded optionsthat protect the policyholder’s investment and/or stream of annuity payments are offered, in exchangefor additional fees, when a variable annuity policy is sold. Common types of options/guarantees includeGuaranteed Minimum Withdrawal Benefits (GMWBs), Guaranteed Minimum Death Benefits (GMDBs),Guaranteed Minimum Accumulation Benefits (GMABs), and Guaranteed Minimum Income Benefits(GMIBs); Hardy (2003) provides an introduction to these types of guarantees. At the end of the fourthquarter of 2008, U.S. variable annuity net assets totaled $1.1 trillion, and in 2008 total sales were$155 billion (NAVA 2009). Clearly, understanding the value of and risks associated with variable annuityguarantees is of significant financial importance. A field of research related to the pricing of these

* Claymore Marshall, AIAA, is a PhD candidate in the Department of Statistics and Actuarial Science at the University of Waterloo, 200 UniversityAvenue West, Waterloo, ON N2L 3G1, Canada, [email protected].† Mary Hardy, FSA, FIA, CERA, PhD, is the CIBC Professor of Financial Risk Management in the Department of Statistics and Actuarial Scienceat the University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada, [email protected].‡ David Saunders, PhD, is an Assistant Professor in the Department of Statistics and Actuarial Science at the University of Waterloo, 200 UniversityAvenue West, Waterloo, ON N2L 3G1, Canada, [email protected].

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 39

guarantees is emerging. Several authors have priced the popular GMWB, all suggesting that insurancecompanies are underpricing this guarantee (Milevsky and Salisbury 2006; Chen et al. 2008; Chen andForsyth 2008; Dai et al. 2008). Literature on GMDBs includes Milevsky and Posner (2001) and Ulm(2009).

There seems to be little in the academic literature discussing GMIBs. In Europe there is a variableannuity guarantee, referred to as a guaranteed annuity option, that has features that are similar to aGMIB, namely, a guaranteed minimum payment rate. Boyle and Hardy (2003), Ballotta and Haberman(2006), and Pelsser (2003) discuss the pricing and hedging of guaranteed annuity options. However,guaranteed annuity options have a different benefit structure to GMIBs at annuitization; hence findingson guaranteed annuity options cannot be directly applied to GMIBs. Bauer et al. (2008) propose auniversal pricing framework for guaranteed minimum benefits in variable annuities, presenting nu-merical results for the GMDB, GMAB, GMWB, and GMIB, all based on a model in which the investmentaccount is modeled as a geometric Brownian motion. In this paper we consider the valuation of a GMIBin more detail and focus on the design elements of GMIBs. The model we use for valuation is anextension of that in Bauer et al., as we allow interest rates to follow a random process. Given that theaccumulation phase must exceed a decade as part of the contract requirements for a GMIB, incorpo-rating a stochastic interest rate model seems worthwhile. Bauer et al. also assume the fee rate chargedfor the GMIB option is a percentage of the investment account, but the GMIB fee structure in practicein the United States is often a fixed percentage of the so-called benefit base rather than the investmentaccount (see eq. [3.3] below), and we adopt this common fee structure in our valuations. Assumingthe same fee rate is applied to the investment account and the benefit base, the fees charged basedon the benefit base are always at least as large as those based on the investment account. It is notedthat Bauer et al. present results on the impact of the inclusion of a GMDB with a GMIB, whereas weconsider the valuation of the GMIB in isolation. We present a comprehensive sensitivity analysis of ourmodel parameters. We decompose the GMIB payoff and measure the contribution to the total GMIBvalue from each of the benefits that are provided by the option. We conclude that GMIBs appear to beunderpriced by insurance companies, which agrees with the existing GMIB pricing results of Bauer etal. It seems that the fair fee rates we obtain are higher than those reported by Bauer et al. However,Bauer et al. present the fair fee rates that should be charged for each of the individual benefit com-ponents provided by a GMIB but not for the GMIB as a whole, whereas we calculate the fair fee ratesfor the GMIB as a whole.

In this paper we value a GMIB in a complete market and find the fair fee rate that should be charged.The numerical results presented in this paper provide a benchmark for GMIB valuations that use moresophisticated models and complex assumptions. The value of the GMIB is broken up into pieces, whereeach piece represents the value of one of the benefits provided by the GMIB. The techniques we useto value the GMIB are simple, but they are effective at generating meaningful information to GMIBsellers (such as whether the fee rates they are charging for the GMIB in practice make sense in a highlysimplified model of reality), and they act as a guide as to things that can be done by insurance com-panies when they are valuing and monitoring the risks associated with products involving similar com-plex financial guarantees/options.

The remainder of this paper is structured as follows. Section 2 discusses the specific nature of thepayoff of a GMIB and lists the assumptions we adopt to value a GMIB. Section 3 discusses the modelused to value a GMIB. In Section 4 we determine the fair fee rate that should be charged for a GMIBand provide a sensitivity analysis of the model parameters. Section 5 presents a decomposition of theGMIB payoff and measures the contribution to the total GMIB value from each of the benefits that areprovided by the option. Section 6 gives an indication of the impact of lapses on the value of a GMIB.Section 7 presents concluding remarks.

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40 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

2. GMIB PAYOFF

Insurance companies offer different bells and whistles with their GMIBs, but the core benefits areessentially the same. In practice, at annuitization a variable annuity with a GMIB gives the policyholderthree options: the policyholder can choose to receive back the accumulated value of the investmentaccount, annuitize the accumulated funds at variable annuity payment rates applicable at the time, orannuitize a guaranteed amount of funds at a payment rate g per annum. Based on publicly availableinformation (AnnuityFYI 2009), the payoff of a variable annuity with a GMIB at the date of annuitizationT is

Y(T) � max{BB(T)ga(T), S (T)}, (2.1)f

where

• Sf(T) is the value of the policyholder’s investment account at time T. The insurance company recoversthe cost of providing the GMIB by charging periodic fees against the investment account. The sub-script f indicates that the investment account is periodically reduced by fees.

• BB(T) is the benefit base of the GMIB, defined asTBB(T) � max{S(0)(1 � r ) , max S (n)}, n � 1, 2, . . . , T, (2.2)g f

n

where S(0) is the initial premium and rg is a guaranteed annual rate that is typically set somewherein the range of 4–6% per annum.

• a(T) is the market price of an annuity with payments of $1 per annum beginning at time T.• g is the guaranteed (annual) annuity payment rate specified by the insurance company at the outset

of the variable annuity contract. It is set conservatively with respect to future mortality and interestrate assumptions.

Equations (2.1) and (2.2) combined illustrate the attractiveness of a GMIB to variable annuity buyers.If investment returns are strong during the accumulation phase, then the policyholder is likely toannuitize the investment at annuity payment rates available at time T (i.e., receive the payoff Sf(T)).However, if investment returns are poor, then the policyholder is able to convert a guaranteed minimumamount of funds—the benefit base BB(T)—into an annuity at a payment rate of g per annum. Thebenefit base provides a guaranteed return of rg per annum; many of the insurance companies sellingGMIBs are currently offering a guaranteed return of 5%. The benefit base also provides the right toreceive the maximum of the investment account on any previous policy anniversary,1 giving the policy-holder the opportunity to lock in gains when investment returns are strong during the accumulationphase.

There are several points to note regarding GMIBs:

• The benefit base is used only for calculating the guaranteed minimum payments and cannot bewithdrawn as a cash lump sum.

• At annuitization even if BB(T) � Sf(T), the policyholder may still be better off annuitizing theiraccumulated investment at the current annuity payment rates rather than exercising the GMIB optionbecause insurance companies set g conservatively.

• For brevity, in the remainder of this paper the two components that form the benefits of a GMIB inthe payoff given by equation (2.1), S(0)(1 � rg)Tga(T) and maxn Sf(n)ga(T), are referred to as theguaranteed return component and the maximum component, respectively.

• The value of g, which is set by the insurance company at the outset, has a large influence on thevalue of a GMIB.

1 The sampling of the maximum of the investment account could be made at any frequency (quarterly, monthly, continuous), but we choosethe sampling frequency to be annual, which is the most common sampling frequency offered in practice.

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 41

• When a GMIB is included in a variable annuity contract, the policyholder cannot annuitize the con-tract for at least the first 10 years from inception of the contract, unless he or she is willing to forfeitthe GMIB option. In practice, the policyholder may annuitize with the guarantee at any point afterT � 10, but in this paper the annuitization date is fixed at the 10th policy anniversary. However, theinvestigation of the optimal annuitization date is an interesting topic for future research.

The following assumptions are made when valuing a GMIB:

• The policyholder chooses to receive a 20-year term certain variable annuity. This assumption isadopted in order to price a GMIB in a complete market, and the justification for it is discussed inSection 3. However, many policyholders are likely to choose a life-related annuity, and thus longevityrisk is an important consideration.

• The policyholder does not lapse before the annuitization date. In practice, if a policyholder lapses,then the GMIB option is forfeited and the insurance company keeps all the GMIB fees earned. Thisis a strong assumption and is financially significant, partly because the policyholder may need towithdraw their invested funds for some unforeseen reason before the minimum 10 years that arerequired for the policyholder to be entitled to the rights provided by the GMIB. Allowing for lapseswill decrease the GMIB value. However, lapses are notoriously difficult to predict, as there are littledata, and because they tend to be correlated to the prevailing economic conditions. Notwithstanding,in Section 6 the effect on the GMIB value from assuming a constant annual lapse rate is explored.

• The policyholder pays a single premium at inception of the policy and does not make any cashwithdrawals. In practice the policyholder may be able to pay additional premiums (which are coveredby the guarantee 10 years later) and make withdrawals (for example, up to a certain percentage ofthe benefit base each year without penalty).

• Some GMIBs give the policyholder the right to ‘‘reset’’ the guaranteed component of the benefitbase at the end of the tth policy year, setting S(0)(1 � rg)t to Sf(t) in the benefit base, which continuesto earn a guaranteed annual rate of rg. However, the policyholder must wait at least another 10 yearsbefore annuitization is possible. This additional complexity is not considered, but such a feature onlyfurther increases the GMIB value.

• Annuity payments are often paid monthly, but we assume payments occur annually in advance. Thisassumption will not lead to a significant difference in the GMIB value.

• Administrative and investment management fees associated with the underlying variable annuity con-tract are not incorporated into the valuation. The actual size of these fees can be somewhere between0.5% and 3% per annum of the policyholder’s investment account during the accumulation phase.Although the impact of these fees on the payoff at time T is not negligible, the actual size of thesefees varies with insurance company, the policyholder’s choice of investment manager, and mortalityassumptions (which we have not considered), and thus making allowances for these fees is rathersubjective. Our interest is in determining a fair fee rate for the GMIB option. It is noted that incor-porating such fees into the methodology presented in this paper can be easily done if the sizes ofthese fees are known with reasonable certainty.

3. VALUATION MODEL

This section discusses the model used to value a GMIB. In this paper we analyze the value of a GMIBon a 20-year term certain annuity with annual payments in advance, with annuitization at 10 yearsfrom inception of the policy. Thus, the analysis focusses on the financial risks, and mortality assump-tions are ignored. By introducing a life-related annuity, we are no longer pricing in a complete market,and the theory for pricing becomes complicated. Currently there are no actively traded financial in-struments that can be used to hedge the longevity component of a GMIB associated with a life annuity;it seems highly unlikely to be able to construct a replicating portfolio in practice. However, it is notedthat longevity risk, a nondiversifiable risk, is an important consideration for lifetime annuities, and themortality assumptions employed would be a key driver of the cost of GMIBs on such annuities.

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42 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

One justification for assuming that the underlying annuity is a 20-year term certain annuity is thatat age 65 the life expectancies for males and females in the 2005 U.S. period life table are 16.7 and19.5 years, respectively (Social Security Online 2009), and age 65 is a likely retirement age for manyvariable annuity buyers. Buying a 20-year term certain annuity will cover the expected number ofpayments that they will need while they are alive. Another justification is that the term certain annuityis also actually one of the choices of annuity type that a variable annuity policyholder may choose inpractice, where the term may be 20 to 30 years.

We value the GMIB using the no-arbitrage pricing methodology. In other words, a replicating port-folio exists that matches the payoff of the GMIB exactly, and we are calculating the value of thereplicating portfolio.

Changes in the value of the investment account between fee payment dates are modeled under therisk-neutral measure by the stochastic differential equation (SDE)

QdS(t) � r(t)S(t) dt � � S(t) dW (t), (3.1)S S

where S(t) is the value of the investment account at time t, r(t) is the short rate at time t, �S � 0 isthe (annualized) instantaneous volatility of the investment account, and { (t), t � [0, T]} is a stan-QW S

dard Brownian motion under the risk-neutral measure Q.The Hull-White (extended Vasicek) model is employed for the term structure of interest rates.

Namely, the instantaneous short rate is modeled under the risk-neutral measure by the SDEQdr(t) � �{�(t)/� � r(t)} dt � � dW (t), (3.2)r r

where � � 0 is a constant, �(t) is a deterministic function of time that depends on the real-world zero-coupon bond yield curve (Bjork [2004] gives its specification), �r � 0 is the instantaneous volatilityof the short rate, and { (t), t � [0, T]} is a standard Brownian motion under Q that is independentQW r

of { (t), t � [0,T]}. Note that �(t) is also a function of �r and �.QW S

Clearly, more sophisticated models for the underlying processes are available, such as investmentaccount processes that allow for jumps and multifactor interest rate processes. However, the mainmotivation for our choice is the fact that the models we employ are well-understood benchmarks, andtheir use allows us to isolate and focus on the influence of the contract features rather than idiosyn-crasies of the assumed processes.

The insurance company does not receive an option premium at the outset for providing a GMIBoption but deducts fees annually from the investment account. The size of the fee charged at the endof policy year n is

�(n) � cBB(n), n � 1, 2, . . . , T, (3.3)

where c � 0 and BB(n) is defined in equation (2.2). Here �(n) is always greater than or equal tocSf(n). Currently in practice c is in the range of 0.5–1% per annum (AnnuityFYI 2009).

Let

TQV(c) � E exp � � r(t) dt max{BB(T)ga (T), S (T)} (3.4)� � � �20 f

0

denote the GMIB value when the fee rate charged is c. In equation (3.4) Q denotes the risk-neutralmeasure and a20(T) is the value of a 20-year term certain annuity at time T.

If the issuer wants to hedge the GMIB by investing in the replicating portfolio, the fee rate c � c*is fair if

V(c*) � S(0). (3.5)

As the fee rate c increases, the GMIB value V(c) decreases, but there is a fee rate threshold beyondwhich increasing the fee rate no longer reduces the GMIB value. This occurs because the guaranteedreturn component of the benefit base is independent of the fee rate. It is possible to value a20(T) by

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 43

Table 1Fair Values of g for a 20-Year Term Certain Annuity

Interest rate 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Annuity value 20.0 18.2 16.7 15.3 14.1 13.1 12.2 11.3 10.6 10.0Estimate of g 0.050 0.055 0.060 0.065 0.071 0.076 0.082 0.088 0.094 0.101

computing the zero-coupon bond prices at time T that are explicitly known for the Hull-White model(Bjork 2004). In equation (3.4), a20(T) � p(T, T � i), where p(T, T � i) is the zero-coupon bond19�i�0

price calculated at time T with maturity date T � i, conditional on r(T).In understanding equation (3.5) it is useful to recall the justification for the arbitrage-free price of

the payoff on the left-hand side being equal to S(0) when c is equal to the fair fee rate. Recall that themarket is assumed to be complete. From derivative pricing theory (Bjork 2004; Hull 2008) it is knownthat the payoff is replicated exactly by investing in a portfolio consisting of the stock index and a zerocoupon bonds costing S(0) dollars at time 0, and then rebalancing that portfolio dynamically in a self-financing way until time T. More succinctly, the payoff of the derivative can be reproduced exactly byinvesting S(0) dollars at time 0 and following a predefined replicating strategy. Note that this repli-cating strategy is distinct from the concept of investing the policyholder’s premium in the mutual fundof their choice at the outset, and then periodically withdrawing fees from the policyholder’s investment.

If the equality in equation (3.4) does not hold, then the insurer has made either a profit or loss atthe inception of the policy. Let P(c) denote the cost of hedging when the fee rate charged is c. Thenin a complete market

P(c) � V(c) � S(0). (3.6)

If P(c) is positive (negative), then the insurance company is undercharging (overcharging) the policy-holder for the GMIB.

4. VALUATION RESULTS AND SENSITIVITY ANALYSIS

In this section we display the GMIB value as a function of the fee rate and determine the fair fee rate,for a realistic range of values of g. The sensitivity of the model to our choice of parameter values forthe underlying processes is then explored. Because of the complexity of the GMIB payoff, Monte Carlosimulation is employed for pricing the GMIB. One hundred thousand simulations are used for all resultspresented in this paper. The following parameter values are used, unless indicated otherwise: S(0) �1,000, T � 10, rg � 5%, �s � 20%, a � 0.35, �r � 1.5%, and �(t) depends on a linear approximationof the shape of the U.S. zero-coupon bond yield curve halfway through 2008 (the curve is displayedlater in Fig. 5 as the one labeled ‘‘standard’’).

4.1 Choice of gThe benefits provided by a GMIB are proportional to the value of g. Hence, the value of g is veryimportant when determining the GMIB value. The insurance company chooses the value of the guar-anteed payment rate g at the inception of the contract. The value of g must be equitable and compet-itive and depends on the type of annuity to which the GMIB is added. A 20-year term certain annuityg is likely to be in the range 5–10%. The justification for this range is as follows: if g is set fairly, thenit should be approximately equal to the inverse of the value of a 20-year term certain annuity. Thevalue of the annuity depends on assumed interest rates over the next 20 years. Table 1 displays valuesof 20-year term certain annuities (with annual payments made in advance) for various interest rates,assuming the interest rate term structure remains flat and constant. The estimates of g in the tableare the inverses of the annuity values. Values of g between 5% and 10% lie in a realistic range of constantinterest rate term structures over the next 20 years. If interest rates do not exceed, say, 9% over the

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44 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

Figure 1GMIB value V(c) as a function of the fee rate c (per $1,000 premium). Each curve corresponds toa particular value of g. For curves that intersect with the horizontal dotted line, the fee rate at

the intersecting point corresponds to the fair fee rate.

0 0.02 0.04 0.06 0.08 0.1600

800

1000

1200

1400

1600

1800

fee rate c

GM

IB v

alue

V(c

)

5%

10%

9%

9.5%

5.5%

7.5%

8.5%

8%

6.5%

7%

6%

long term (plausible, based on recent history in the United States), then a competitive/equitable valuefor g seems to be somewhere between 5% and 8%. In making our choice of g for valuation purposes,we keep in mind that insurance companies set g conservatively, and that when the GMIB option is‘‘exercised,’’ the amount of funds annuitized is the benefit base, BB(T), which is at least as large asthe investment account, Sf(T) (implying the g values in the table are not, in a sense, the fair values).Hence in this paper we present numerical results typically for g of 5.5%, 6.5%, 7.5%, and 8.5%, whicharguably correspond to conservative through to optimistic assumptions, from the insurance company’sperspective, for funding the guarantees. In this paper we consider a value of g of 6.5–7.5% to be about‘‘average’’ and equitable for the policyholder.

4.2 The Fair Fee RateFigure 1 illustrates the relationship between the GMIB value V(c) and the fee rate c for a wide rangeof values of g. (The standard errors of the GMIB value estimates lie in the range $0.4–2.4.) Each curvecorresponds to the GMIB value for a given g, and the fee rate at the intersecting point of a curve withthe horizontal dotted line corresponds to the fair fee rate for the curve. For any given fee rate, thevertical distance between a curve and the horizontal dotted line corresponds to the cost of hedging.When any of the curves lie below the horizontal dotted line, the cost of hedging is negative, and it canbe loosely thought of as profit for the insurer. The fair fee rates for g of 5%, 5.5%, 6%, 6.5%, and 7%are 1.2%, 1.85%, 2.85%, 4.65%, and 9.25%, respectively. For g greater than 7% it is not possible for theinsurance company to break even (based on our model parameter assumptions); hedging the GMIBusing a replicating portfolio requires the insurer to obtain funds from elsewhere. When g � 7%, theinitial investment S(0) cannot ever cover the cost of the guarantee because the guaranteed returncomponent of the GMIB, S(0)(1 � rg)Tga20(T), is too valuable and is independent of the fee rate. Notethat currently in practice insurance companies are charging fees of 0.5–1% for GMIB options; seeAnnuityFYI (2009) for a summary of the fees charged by several major U.S. providers. This simplemodel suggests that insurance companies may be underpricing GMIBs for average/equitable values of

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 45

Figure 2Left panel displays the GMIB value without fee charges V(0) as a function of investment accountvolatility �S. Right panel displays the GMIB value as a function of investment account volatility

when the fair fee rate is charged. Each curve corresponds to a particular value of g.

0 0.1 0.2 0.3 0.4 0.51000

1200

1400

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investment account volatility σS

GM

IB v

alue

V(0

)

g=5.5%g=6.5%g=7.5%g=8.5%

0 0.1 0.2 0.3 0.4 0.5800

900

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IB v

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) us

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rate

g=5%g=5.5%g=6%g=6.5%

g. However, the prices presented ignore lapse assumptions. Given that the policyholder must not with-draw any funds until T � 10—if the policyholder wants to have the GMIB option at annuitization—itseems probable that there would be some policy lapses, in which case the fair fee rate would be reduced;Section 6 examines this issue. Furthermore, as mentioned in Section 2, our analysis has not allowedfor fees relating to the underlying variable annuity contract, which can be 0.5–3% per annum of theinvestment account.

The fair fee rates we have obtained for a GMIB seem to be slightly higher than the fair fee ratesreported in Bauer et al. (2008) for GMIBs. However, Bauer et al. value each of the benefits providedby a GMIB in isolation and determine the fair fee rates for each individual benefit but not for the GMIBas a whole. They also use a different fee structure.

4.3 Investment Account VolatilityThe left panel of Figure 2 displays the relationship between V(0), the GMIB value when the fee rate iszero, and the volatility of the investment account �S using the benchmark values of g. The right panelof Figure 2 shows the GMIB value as a function of the investment account volatility when the fair feerate is charged, for values of g � 7% (fair fee rates do not exist for g � 7%, and the fair fee rate for g� 7% is too high to ever realistically be charged). For example, when g � 6%, the fair fee rate is 1.85%,and the corresponding curve plots the GMIB value using this fee rate as a function of the investmentaccount volatility. (The standard errors of the GMIB value estimates lie in the range $0.2–9.9 andincrease as the volatility increases.) As would be expected, the GMIB value is a monotonically increasingfunction of �S; a higher volatility gives greater probability to larger values of BB(T) and Sf(T) at timeT. In the left panel, starting at a conservative volatility of 10%, each 5% increase in volatility leads toan increase in V(0) of about 4–8%, where the percentage increases in V(0) are gradually increasing.In practice new policyholders are given the choice of investing their money in investment trusts withdifferent risk profiles. Subject to the insurer charging the same fee rate for a given set of trusts, newpolicyholders should invest their money in the trusts with the highest volatilities if they want to max-imize the value of their GMIB.

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46 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

Figure 3Left panel displays the GMIB value without fee charges V(0) as a function of interest rate vlatility�r . Right panel displays the GMIB value as a function of interest rate volatility when the fair fee

rate is charged. Each curve corresponds to a particular value of g.

0 0.02 0.04 0.06 0.08 0.1 0.121100

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interest rate volatility σr

GM

IB v

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V(0

)

g=5.5%g=6.5%g=7.5%g=8.5%

0 0.02 0.04 0.06 0.08 0.1 0.12980

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IB v

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) us

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rate

g=5%g=5.5%g=6%g=6.5%

4.4 Impact of Interest Rate AssumptionsThe Hull-White model is employed for modeling the short rate. In this section we explore the sensitivityof the GMIB value to the parameter values in the Hull-White model. It is shown that the interest rateassumptions have a significant influence on the GMIB value.

The left panel of Figure 3 displays the GMIB value without fee charges V(0) plotted against interestrate volatility �r using the benchmark values of g. The right panel of Figure 3 displays the GMIB valueas a function of interest rate volatility when the fair fee rate is charged, for lower values of g where thefair fee rate exists. (The standard errors of the GMIB value estimates lie in the range $0.9–5.2.) TheGMIB value is a monotonically increasing function of �r. Because the GMIB option is sensitive tointerest rates through the annuity factor, it is intuitive that its value should increase with interest ratevolatility. Note that a higher interest rate volatility leads to greater variability in the discounting factorand in the drift of the investment account SDE. In the left panel, each 1% increase in interest ratevolatility leads to an increase in V(0) of about 0.5–2%, where the percentage increases in V(0) aregradually increasing. Changes in interest rate volatility have a much smaller influence on the GMIBvalue compared to changes in investment account volatility. For interest rate volatilities of less than2%, which are arguably realistic values for the past decade, the GMIB value remains fairly constant.

The left panel of Figure 4 shows the relationship between the GMIB value without fees V(0) and theparameter � in the SDE given by equation (3.2), using the benchmark values of g. The right panel ofFigure 4 displays the GMIB value as a function of � when the fair fee rate is charged, for lower valuesof g where the fair fee rate exists. (The standard errors of the GMIB value estimates lie in the range$1.7–4.1.) We note that in equation (3.4) a20(T) � p(T, T � i) is a function of �. For our calibration19�i�0

of the yield curve, captured in �(t), the GMIB value is relatively insensitive to the value of �.Figure 5 displays five different zero-coupon bond yield curves used for testing the sensitivity of the

GMIB value to the underlying assumed yield curve in the Hull-White model (the yield curve shapeaffects �(t) in eq. [3.2]). The curve labeled ‘‘Benchmark’’ is the curve applied to all of the valuationspresented in this paper unless stated otherwise; it is a linear approximation of the shape of the U.S.zero-coupon bond yield curve halfway through 2008. The curves labeled ‘‘3% Shift’’ and ‘‘6% Shift’’ areparallel upward shifts of the Benchmark curve, where the sizes of the shifts are 3% and 6%, respectively.The shifted curves could occur in practice under different economic conditions to the present (e.g.,

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 47

Figure 4Left panel displays the GMIB value without fee charges V(0) as a function of (Hull-White

parameter) �. Right panel displays the GMIB value as a function of � when the fair fee rate ischarged. Each curve corresponds to a particular value of g.

0.5 1 1.5 2 2.5 31100

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IB v

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)

g=5.5%g=6.5%g=7.5%g=8.5%

0 0.5 1 1.5 2 2.5 3980

985

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1015

α

GM

IB v

alue

V(c

) us

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fee

rate

g=5%g=5.5%g=6%g=6.5%

Figure 5Set of zero-coupon bond yield curves used for testing the sensitivity of GMIB value to the

underlying assumed yield curve. Figure 6 shows the corresponding GMIB values.

0 5 10 15 20 25 300.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

year

zero

cou

pon

bond

yie

ld

Benchmark3% Shift6% ShiftChange in ConvexityInverse

when inflation is high). The curve labeled ‘‘Change in Convexity’’ represents a change in convexity ofthe Benchmark curve. The shape of the Change in Convexity curve is convex rather than concave asfor the Benchmark curve, but to facilitate a comparison with the Benchmark curve the level of theChange in Convexity curve is roughly the same as the Benchmark curve at the short and long maturitydates. Finally, the curve labeled ‘‘Inverse’’ captures the shape of an inverted yield curve, where the levelof this curve at the short maturity dates is kept close to the Benchmark curve.

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48 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

Figure 6GMIB value V(c) as a function of fee rate c, assuming g � 6.5% for each curve in left panel and

g � 7.5% for each curve in right panel. Each curve plots the GMIB value using the correspondingyield curve in Figure 5.

0 0.02 0.04 0.06 0.08 0.1400

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fee rate c

GM

IB v

alue

V(c

)

Benchmark3% Shift6% ShiftChange in ConvexityInverse

g=6.5%

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fee rate c

GM

IB v

alue

V(c

)

Benchmark3% Shift6% ShiftChange in ConvexityInverse

g=7.5%

The panels in Figure 6 display the GMIB value as a function of the fee rate for each of the yieldcurves in Figure 5, for g � 6.5% and 7.5%. Each curve corresponds to the GMIB value for a given yieldcurve. (The standard errors of the GMIB value estimates lie in the range $0.5–3.2.) All else beingconstant, as the level of the yield curve increases, the GMIB value decreases; the 3% Shift curve isuniformly lower than the Benchmark curve, and similarly the 6% Shift curve is uniformly lower thanthe 3% Shift curve. This observation is intuitive, but it is important to realize that several factors affectthe GMIB value in opposite directions when the yield curve is shifted:

1. The short rate reverts to �(t)/�, and �(t) is larger for all t when the yield curve is shifted upwards.Thus the discounting factor will be larger, reducing the GMIB value.

2. A higher yield curve reduces a20(T), scaling down the payoffs of the maximum and guaranteed returncomponents. This also reduces the GMIB value.

3. The drift coefficient in the SDE of the investment account depends on the short rate, and the shortrate will tend to follow higher paths during the accumulation phase when �(t) is larger for all t.Hence the investment account will also tend to follow higher paths during the accumulation phase,increasing the GMIB value.

Figure 6 shows that the effects of (1) and (2) overwhelm the effect of (3). Figure 7, discussed below,also provides further indication that as interest rates increase, the GMIB value decreases.

By comparing the Change in Convexity and Benchmark curves in Figure 6, it is clear that the con-vexity of the yield curve has a significant impact on the GMIB value. Figure 5 shows that the Changein Convexity curve is lower than the Benchmark curve for all but the longest maturities; this is thereason why in Figure 6 the Change in Convexity curve is higher than the Benchmark curve. The Inversecurve in Figure 6 is the highest of all the curves, demonstrating that the level of the long end of theyield curve significantly affects the GMIB value. In Figure 5 the yields at the long end of the Inversecurve are the lowest among all the curves. Hence if the long end of a yield curve decreases, then theGMIB value will be driven up significantly. The GMIB value increases partly because if the long endfalls, then bond prices with maturity dates beyond time T increase, increasing the value of a20(T). Theother cause of the increase in value is that the discounting factor is also lower because the short ratepaths tend to be lower during the accumulation phase.

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 49

Figure 7Relationship between GMIB value V(c) and fee rate c for various constant interest rates r,

assuming g � 6.5% for each curve in left panel and g � 7.5% for each curve in right panel. Eachcurve corresponds to a particular value of r.

0 0.02 0.04 0.06 0.08 0.1700

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fee rate c

GM

IB v

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)

r=2%r=3%r=4%r=5%r=6%

g=6.5%

0 0.02 0.04 0.06 0.08 0.1800

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fee rate c

GM

IB v

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V(c

)

r=2%r=3%r=4%r=5%r=6%

g=7.5%

It is clear that assumptions for the underlying yield curve shape have a large influence on the valueof the GMIB, largely because of the long time to expiration of the payoff. Before moving on to otherissues we consider the impact on the GMIB value from removing the complication of variable interestrates. The panels in Figure 7 display the GMIB value V(c) as a function of the fee rate c for g � 6.5%and 7.5%. Each curve assumes the term structure of interest rates is flat and constant through timeat a particular rate r; r takes the values 2%, 3%, 4%, 5%, and 6%. (The standard errors of the GMIBvalue estimates lie in the range $0.3–2.1.) Clearly, when the term structure is flat and shifted upwards,the GMIB value decreases. As the interest rate r increases, the discounting factor and the annuity valueboth decrease in value, reducing the GMIB value. The outcome of effects (1), (2), and (3) is againdemonstrated in Figure 7, where a higher interest rate r corresponds to shifting up the yield curvestructure. Each 1% increase in the interest rate r leads to a decrease in the GMIB value V(c) of about6–11%. The results in Figure 7 can be loosely compared to those of Bauer et al. (2008) because theshort rate is deterministic, although they use different values for T and �S, and set rg � 6%, which usedto be a very common guaranteed rate in practice until the recent financial crisis occurred. Bauer etal. (2008) report that, using r � 4%, no fair fee rate exists for a GMIB when g is above a certain value.2

In the left panel of Figure 7, where g � 6.5%, the fair fee rate for r � 4% does not exist, which agreeswith the results of Bauer et al.

4.5 Correlation between Underlying ProcessesThe results presented thus far have assumed that the short rate and investment account processesevolve independently over time. This section considers the impact on the GMIB value when theseprocesses are correlated. Let � be the linear correlation coefficient between { (t), t � [0,T]} andQW S

{ (t), t � [0,T]} such thatQW r

� dt if t � u,Q QCov(dW (t), dW (u)) � �S r 0 if t � u.

2 Bauer et al. (2008) show there is no fair fee rate for what we call the guaranteed return component of the GMIB, not the ‘‘entire’’ GMIBcontract, but the implications are the same.

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50 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

Figure 8Relationship between GMIB value V(c) and fee rate c for various values of �, assuming g � 6.5%for each curve in left panel and g � 7.5% for each curve in right panel. Each curve corresponds

to a particular value of �.

0 0.02 0.04 0.06 0.08 0.1900

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GM

IB v

alue

V(c

)

ρ=+1.0

ρ=+0.5

ρ=0

ρ=0.5

ρ=–1.0

g=6.5%

0 0.02 0.04 0.06 0.08 0.11050

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fee rate cG

MIB

val

ue V

(c)

ρ=+1.0ρ=+0.5ρ=0ρ=0.5ρ=–1.0

g=7.5%

The panels in Figure 8 compare the GMIB value V(c) as a function of the fee rate c for various valuesof �, for g � 6.5% and 7.5%. (The standard errors of the GMIB value estimates lie in the range $0.6–1.9.) It appears that the GMIB value is a monotone increasing function of �. It is noted that the samepatterns emerge for other values of g. It is difficult to give a clear explanation for this observed behavior,because regardless of whether the correlation is positive or negative, there are always multiple effectsthat influence the GMIB value in opposite directions. For example, if the short rate tends to increasewhen the investment account decreases, the drift of the investment account SDE will also increase;thus the overall change in the value of the investment account is not obvious. However, a partialexplanation for the observed behavior is suggested: If the correlation is positive, then the Brownianmotion components in the SDEs of both processes are both likely to move in the same direction, andin the investment account SDE (given by eq. [3]), there is a magnifying effect on the movement of theinvestment account because the drift term also moves in the same direction as the random component.This compounding effect causes the investment account volatility to be slightly higher, and Section4.3 has already indicated that the GMIB value is sensitive to the investment account volatility. If thecorrelation is negative, then the opposite effect occurs.

4.6 Varying the GMIB Benefit ParametersThe insurance company must decide what guaranteed rate rg it will provide with its GMIB. Currently(end of 2008) rg is typically somewhere between 4% and 6%, although 5% seems to be common and isused in the valuations presented in this paper. The left panel of Figure 9 displays the GMIB valuewithout fees V(0) as a function of the guaranteed rate of return rg for the benchmark values of g. Theright panel of Figure 9 displays the GMIB value as a function of rg when the fair fee rate is charged,for lower values of g where the fair fee rate exists. (The standard errors of the GMIB value estimateslie in the range $1.7–2.2.) The GMIB value increases monotonically with rg. In the left panel, each0.5% increase in the guaranteed rate of return increases V(0) by 1–2% if g is 5.5%, 1.5–3% if g is 6.5%or 7.5%, and 2–3.5% if g is 8.5%, where the percentage increases in V(0) are gradually increasing.

In this paper the simplifying assumption that the policyholder annuitizes at T � 10 years is adopted.In reality the policyholder may choose to annuitize at any point after 10 years. The panels in Figure10 plot the GMIB value V(c) as a function of the fee rate c for T of 10, 20, and 30 years, using g �6.5% and 7.5%. (The standard errors of the GMIB value estimates when T is 10, 20, and 30 lie in the

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 51

Figure 9Left panel displays the GMIB value without fees V(0) as a function of guaranteed annual rate of

return rg. Right panel displays the GMIB value as a function of rg when the fair fee rate ischarged. Each curve corresponds to a particular value of g.

0.02 0.03 0.04 0.05 0.06 0.07 0.081000

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Figure 10Relationship between GMIB value V(c) and fee rate c for T � 10, 20, and 30 assuming g � 6.5%in left panel and g � 7.5% in right panel. Each curve corresponds to a particular annuitization

date T.

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ranges $0.7–1.8, $0.5–3.3, and $0.6–4.9, respectively.) The zero-coupon bond yield curve used in theshort rate model is the Benchmark curve up to 30 years, and then from 30 to 50 years the zero-couponbond yield curve is assumed to increase linearly from 4.88% to 5%. As T increases, BB(T) increases invalue (increasing the GMIB value), the discounting factor decreases in value (decreasing the GMIBvalue), and a larger number of (annual) fee deductions from the investment account are made (de-creasing the GMIB value). Figure 10 suggests that there is no clear relationship between increasingthe time until annuitization and the GMIB value—the GMIB value may increase or decrease dependingon the fee rate charged.

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52 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

5. DECOMPOSING THE GMIB VALUE

This section explores why the GMIB appears to be quite valuable. The GMIB payoff is decomposed,facilitating an understanding of the drivers of its value. This analysis also provides useful informationfor risk management purposes.

5.1 Contributions of Each Benefit to the GMIB ValueIn this section we measure the contribution of each benefit component of a GMIB (as given by eq.[2.1] and [2.2]) to the overall GMIB value. This concept is important from hedging, risk management,and (future) product design perspectives. At the annuitization date T, the GMIB payoff is the maximumof three possible outcomes:

TX � max S (n)ga (T), X � S(0)(1 � r ) ga (T), X � S (T).1 f 20 2 g 20 3 fn

Let

T

Y � exp � � r(t) dt X , i � 1, 2, 3.� �i i0

The contribution to the GMIB value V(c) of each outcome can be obtained by reexpressing the GMIBvalue as the sum of three components:

Q Q QV(c) � E [Y I ] � E [Y I ] � E [Y I ], (5.1)1 {X �X ,X } 2 {X �X ,X } 3 {X �X ,X }1 2 3 2 1 3 3 1 2

where I is the indicator function.3

Let

Qy � E [Y I ] (contribution from the maximum component),1 1 {X �X ,X }1 2 3

Qy � E [Y I ] (contribution from the guaranteed return component),2 2 {X �X ,X }2 1 3

Qy � E [Y I ] (contribution from the investment account).3 3 {X �X ,X }3 1 2

The panels in Figure 11 display yi, i � 1, 2, 3 as functions of the fee rate for g of 5.5%, 6.5%, 7.5%,and 8.5%. (The standard errors of the yi estimates lie in the range $0.5–4.2.) In each panel, the sumof the values of the three curves for any given fee rate equals the GMIB value (at that fee rate). As thefee rate increases, the sum of the three curves must decrease (to a lower bound, as seen in Fig. 1).

In the top left panel, when g � 5.5%, it is clear that most of the GMIB value comes from thecontribution from the investment account y3 for fee rates below 2%, and from the contribution of theguaranteed return component y2 for fee rates above 2%. For low fee rates, the contributions of theguaranteed return and maximum components are each worth less than the contribution of the invest-ment account, suggesting the GMIB option is not very valuable when g � 5.5%. However, for high (butunrealistic/unmarketable) fee rates exceeding 5% per annum, the guaranteed return component y2 ofthe GMIB becomes valuable because most of the funds in the investment account are eaten up by highfees. When g � 5.5%, the fair fee rate is 1.85% (see Fig. 1). At this fair fee rate, the investment accountcomponent y3 contributes the most to GMIB value. However, g � 5.5% is fairly conservative, and inpractice g is likely to be higher. The top left panel, which displays the contributions for a more equitableg of 6.5%, indicates that for fee rates above 0.5%, the guaranteed return component y2 contributes themost to the GMIB value. At the fair fee rate of 4.65%, y2 is worth 70% of the GMIB value, while themaximum component y1 is worth 16% of the GMIB value. The bottom panels illustrate that as g getslarger, the maximum component y1 contributes more to the GMIB value than the investment account

3 In Equation (5.1) the cases Xi � Xj, i � j, i, j � 1, 2, 3, have probability 0 and are ignored.

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 53

Figure 11Each panel displays contributions to the GMIB value from yi, i � 1, 2, 3 (the maximum

component, guaranteed return component, and investment account, respectively), as functionsof the fee rate for a particular value of g. Top left panel displays contributions for g � 5.5%, topright panel displays contributions for g � 6.5%, bottom left panel displays contributions for g �

7.5%, and bottom right panel displays contributions for g � 8.5%.

0 0.02 0.04 0.06 0.08 0.10

100

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fee rate c

risk-

neut

ral e

xpec

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pres

ent v

alue

y1 (maximum)

y2 (guaranteed r

g)

y3 (inv. account)

g=5.5%

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

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600

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800

900

fee rate c

risk-

neut

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xpec

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pres

ent v

alue

y1 (maximum)

y2 (guaranteed r

g)

y3 (inv. account)

g=6.5%

0 0.02 0.04 0.06 0.08 0.10

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fee rate c

risk-

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y1 (maximum)

y2 (guaranteed r

g)

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g=7.5%

0 0.02 0.04 0.06 0.08 0.10

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y1 (maximum)

y2 (guaranteed r

g)

y3 (inv. account)

g=8.5%

y3 for any fee rate. As shown in the bottom right panel, when g is sufficiently large, the investmentaccount component y3 has negligible value while the maximum component y1 becomes very valuable.

A number of important observations are deduced from Figure 11:

• It is clear that the guaranteed return component y2 is the dominant contribution to the GMIB valuefor average/equitable values of g. However, when g is sufficiently large, the maximum component y1

becomes at least as valuable as the guaranteed return component y2 at lower fee rates. The investmentaccount y3 is also valuable for conservative values of g.

• As g increases, the contribution of the investment account y3 to the GMIB value decreases sharply.This occurs because as g increases, both the maximum component (X1) and guaranteed return com-ponent (X2) payoffs are scaled up, and thus both are more likely to be worth more relative to thevalue of the investment account.

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54 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

Figure 12Values of EQ[Yi]i � 1, 2, 3 (the maximum component, guaranteed return component, and

investment account, respectively) as functions of the fee rate. Top left panel displays values forg � 5.5%, top right panel displays values for g � 6.5%, bottom left panel displays values for

g � 7.5%, and bottom right panel displays values for g � 8.5%.

0 0.02 0.04 0.06 0.08 0.10

100

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fee rate c

risk-

neut

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pres

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alue

EQ[Y1] (maximum)

EQ[Y2] (guaranteed r

g)

EQ[Y3] (inv. account)

g=5.5%

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fee rate c

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EQ[Y1] (maximum)

EQ[Y2] (guaranteed r

g)

EQ[Y3] (inv. account)

g=6.5%

0 0.02 0.04 0.06 0.08 0.10

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fee rate c

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EQ[Y1] (maximum)

EQ[Y2] (guaranteed r

g)

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g=7.5%

0 0.02 0.04 0.06 0.08 0.10

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EQ[Y2] (guaranteed r

g)

EQ[Y3] (inv. account)

g=8.5%

• As the fee rate increases, the contribution of the guaranteed return component y2 increases whilethe contribution of the maximum component y1 decreases. This is expected because the formercomponent is independent of the fee rate while the latter is a decreasing function of the fee rate.

• The maximum component y1 is less sensitive to the fee rate than the investment account y3, indi-cating that increasing the fee rate reduces the GMIB value primarily through reducing the contri-bution from the investment account y3 rather than the contribution from the maximum componenty1.

A slightly different but closely related perspective as to the drivers of the GMIB value is obtained byvaluing the three outcomes in isolation. The panels in Figure 12 display EQ[Y1] (maximum component),EQ[Y2] (guaranteed return component), and EQ[Y3] (investment account) as functions of the fee ratefor g of 5.5%, 6.5%, 7.5%, and 8.5%. (The standard errors of the EQ[Yi] estimates lie in the range $0.7–2.1.) It is noted that the EQ[Y3] curve is the same in each panel because it does not depend on g. Eachpanel illustrates that when the maximum and guaranteed return components are valued in isolation,

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 55

Figure 13Values of zi, i � 1, 2, 3 (maximum component or investment account, guaranteed return

component or investment account, and GMIB value, respectively) as functions of the fee rate,assuming g � 6.5% in left panel and g � 7.5% in right panel.

0 0.02 0.04 0.06 0.08 0.1600

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z

1 maximum or inv. account

z2 guaranteed r

g or inv. account

z3 GMIB value

g=6.5%

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z2 guaranteed r

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z3 GMIB value

g=7.5%

the latter is more valuable except for very low fee rates. The value of the maximum component decreasesas the fee rate increases, but the rate of decrease becomes smaller as the fee rate increases. Clearlythe observations drawn from Figure 11 are reinforced by Figure 12. For average values of g, the guar-anteed return component is the most valuable part of the GMIB when valuing the three componentsin isolation.

5.2 Valuing Simplified GuaranteesSuppose an insurance company selling GMIBs wanted to offer simpler guarantees for their variableannuities. Specifically, suppose the payoff at annuitization consists of the maximum of the investmentaccount and either the maximum component or the guaranteed return component, but not both. It isuseful to know how much difference there is between the values of these simpler guarantees and theGMIB value. The panels in Figure 13 display the following:

TQz � E exp �� r(t) dt max{X , X } (maximum component or investment account),� � � �1 1 3

0

TQz � E exp �� r(t) dt max{X , X } (guaranteed return component or investment account),� � � �2 2 3

0

TQz � E exp �� r(t) dt max{X , X , X } (GMIB value),� � � �3 1 2 3

0

as functions of the fee rate for g � 6.5% and g � 7.5%. (The standard errors of the zi estimates lie inthe range $0.5–2.8.) A striking observation is that z2 is closer to z3 (the GMIB value) than one mightexpect. However, z3 is substantially larger than z1. This suggests that the maximum component doesnot contribute much to the GMIB value in excess of the guaranteed return component, which is alsosupported by Figure 11. Obviously the inclusion of the maximum component increases the appeal of aGMIB to variable annuity buyers. It might be argued that variable annuity buyers perceive the maximumcomponent to be quite valuable, but, in fact, this guarantee contributes little to the value of a GMIBthat already includes a guaranteed return component. Nevertheless, in spite of the maximum compo-

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56 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 1

Table 2Upfront Fair Fee Rate �* per $1,000 Premium

for Various Values of g

g 5.5% 6.5% 7.5% 8.5% 9.5% 10.5%Fair fee rate 11% 18.5% 26% 33.5% 40.5% 46%

nent appearing to be a cheap benefit for the insurance company to provide, it may still be costly whenit is ‘‘in the money.’’ When the maximum component guarantee is exercised, it may occasionally bevery valuable (if the value of the investment account on any policy anniversary is very high) and thusshould not be ignored when considering hedging strategies for GMIBs. Put another way, a small con-tribution to the overall price does not imply the risk associated with the maximum component guar-antee is also negligible.

5.3 Upfront Fair FeeThe GMIB seller earns the equivalent of an option premium by charging (typically) annual fees duringthe accumulation phase. A simpler but probably somewhat less marketable alternative would be tocharge one upfront fee, with no fees paid thereafter. This section determines the magnitude of such afee. The magnitude of the fee gives another measure of the value of a GMIB, and unlike the annual feepayments approach, a ‘‘fair’’ upfront fee can be calculated for values of g in excess of 7%. Suppose thepolicyholder pays an annuity premium of � at inception (� � S(0) in this paper), and is the upfrontfee rate charged as a percentage of the annuity premium. The insurance company receives a fee of �at the outset and invests �(1 � ) for the policyholder. The upfront fee rate � * is fair if � � V(c;�(1 � *)), where V(�, �(1 � *)) is identical to the function in equation (3.4) but with the initialinvestment S(0) equal to �(1 � *) and c � 0. By charging * the insurance company has the exactamount of funds needed to construct the replicating portfolio for the GMIB. Table 2 shows the fair feerate * for various guaranteed payment rates g. For an equitable value of g in the range of 6.5–7.5%,the fair upfront fee * is about 18.5–25.5% of the annuity premium �. Variable annuity buyers mayfind it far less appealing to pay an upfront fee of this magnitude compared to the alternative of payingsmaller annual fees during the accumulation phase.

6. IMPACT OF LAPSES

Lapses have been ignored in the valuations thus far, but allowing for lapses will reduce the GMIB value.It is dificult to say what an appropriate set of withdrawal rate assumptions during the accumulationphase should be. Hence, in this section we consider the change in the GMIB value from simply assuminga constant annual lapse rate. Let p denote the probability that the policyholder lapses over a givenpolicy year. The GMIB value allowing for lapses is

T TL T Q i�1V (c) � V(c)(1 � p) � E exp �� r(t) dt S (i) (1 � p) p, (6.1) � � � �f

0i�1

where V(c) is still given by equation (3.4). Equation (6.1) assumes that if the policyholder lapses inyear i, they receive the value of the investment account at the end of year i.

The panels in Figure 14 display the GMIB value VL(c) as a function of the fee rate c for various valuesof p, for g � 6.5% and 7.5%. (The standard errors of the VL(c) estimates lie in the range $0.5–1.9.)The curves labeled p � 0% are identical to the curves for g of 6.5% and 7.5% presented in Figure 1.The left panel shows that the fair fee rate drops significantly even for small lapse rates. For example,a conservative lapse rate of 2.5% per annum reduces the fair fee rate to 3.25%. The right panel dem-onstrates that although there is no fair fee rate when g � 7.5% and p � 0, if a small constant lapse

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VALUATION OF A GUARANTEED MINIMUM INCOME BENEFIT 57

Figure 14Relationship between GMIB value VL(c) and fee rate c for various constant lapse rates p, assuming

g � 6.5% for each curve in left panel and g � 7.5% for each curve in right panel. Each curvecorresponds to a particular value of p.

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IB v

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rate is introduced, then it is possible for a fair fee rate to exist. These observations show that the fairfee rate is highly sensitive to lapse assumptions. Hence if policyholder lapse behavior can be reliablymeasured, it should be incorporated into the valuations of GMIBs. A complicating factor is that lapsesdepend on economic conditions in ways that may not be clearly understood, given the short history ofthese products. It is noted that assuming an annual lapse rate as high as 10% still does not reduce thefair fee rate to levels that are currently being charged in practice, when g is set at an average value of6.5%.

7. CONCLUDING REMARKS

This paper has dealt with the valuation of a GMIB associated with a 20-year term certain annuity in acomplete market. The GMIB was valued using straightforward benchmark models, avoiding complexmodels with idiosyncracies. It has been shown that interest rate assumptions have a significant influ-ence on the GMIB value. Our model suggests that, based on reasonable parameter assumptions, thefee rates being charged by insurance companies for GMIBs (currently about 0.5–1% p.a.) may be toolow. Specifically, the GMIB option fee rates being charged are lower than what is needed to hedge theGMIB in a complete market. Of the two guarantees provided by the GMIB, referred to as the guaranteedreturn and maximum components in this paper, the guaranteed return component is the most valuable.Moreover, the maximum component seems to be a relatively cheap guarantee for the insurance com-pany when paired with the guaranteed return component, and potential buyers probably perceive thevalue of this guarantee to be much higher than what has been calculated in this paper.

Assumptions made to simplify the analysis included no policy lapses, no cash withdrawals or addi-tional premiums, no lapses, and the annuitization date being fixed at the 10th policy anniversary.Varying each of these assumptions will lead to changes in the fair fee rates and GMIB values presented.Also, in practice insurance companies might be making profits from selling GMIBs through policyholderlapses because the minimum 10-year accumulation phases may end up being too long for some cash-tight policyholders. Section 6 has shown that the GMIB value is highly sensitive to lapse behavior. Thefair fee rate is significantly dependent on lapse assumptions.

Features of GMIBs worth exploring include assessing the value of the right of the policyholder to‘‘reset’’ the guaranteed component of the benefit before annuitization, measuring the cost of mortality

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improvement for GMIBs associated with life annuities, and exploring the optimal time for annuitizationbeyond 10 years. The challenging issue of hedging GMIBs is an important one that needs investigating,but to date does not seem to have been touched on much in the literature. Investigating the effective-ness of possible static hedges is a first step.

8. ACKNOWLEDGMENTS

The authors express their gratitude to three anonymous referees for their very helpful comments andsuggestions. Claymore Marshall acknowledges generous financial support from the Institute of Actu-aries of Australia (through the 2008 A. H. Pollard Ph.D. Scholarship), the Institute of QuantitativeFinance and Insurance at the University of Waterloo, and Dominion of Canada General Insurance Com-pany during the writing of this paper. He also thanks Janvier Nzeutchap for helpful feedback on thepaper. Mary Hardy and David Saunders acknowledge the support of the Natural Sciences and Engi-neering Research Council of Canada.

REFERENCES

ANNUITYFYI. 2009. Compare Guaranteed Minimum Income Benefit Riders. Available at: www.annuityfyi.com/ca1e living benefit.html.BALLOTTA, L., AND S. HABERMAN. 2006. The Fair Valuation Problem of Guaranteed Annuity Options: The Stochastic Mortality Environ-

ment Case. Insurance: Mathematics and Economics 38: 195–214.BAUER, D., A. KLING, AND J. RUSS. 2008. A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities.

ASTIN Bulletin 38(2): 621–651.T. 2004. Arbitrage Theory in Continuous Time. Oxford: Oxford University Press.BJORK,

BOYLE, P., AND M. R. HARDY. 2003. Guaranteed Annuity Options. ASTIN Bulletin 33(2): 125–152.CHEN, Z., AND P. FORSYTH. 2008. A Numerical Scheme for the Impulse Control Formulation for Pricing Variable Annuities with a

Guaranteed Minimum Withdrawal Benefit (GMWB). Numerische Mathematik 109(4): 535–569.CHEN, Z., K. VETZAL, AND P. A. FORSYTH. 2008. The Effect of Modelling Parameters on the Value of GMWB Guarantees. Insurance:

Mathematics and Economics 43(1): 165–173.DAI, M., Y. K. KWOK, AND J. ZONG. 2008. Guaranteed Minimum Withdrawal Benefit in Variable Annuities. Mathematical Finance 18(4):

595–611.HARDY, M. R. 2003. Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance. New York: John Wiley

& Sons.HULL, J. 2008. Options, Futures and Other Derivatives. 7th ed. New York: Prentice Hall.MILEVSKY, M., AND S. E. POSNER. 2001. The Titanic Option: Valuation of the Guaranteed Minimum Death Benefit in Variable Annuities

and Mutual Funds. Journal of Risk and Insurance 68(1): 91–126.MILEVSKY, M., AND T. S. SALISBURY. 2006. Financial Valuation of Guaranteed Minimum Withdrawal Benefits. Insurance: Mathematics

and Economics 38(1): 21–38.NAVA. 2009. NAVA Reports Fourth Quarter Variable Annuity Industry Data. Available at: www.navanet.org/pressroom/article/id/249.PELSSER, A. 2003. Pricing and Hedging Guaranteed Annuity Options via Static Option Replication. Insurance: Mathematics and

Economics 33(2): 283–296.SOCIAL SECURITY ONLINE. 2009. Actuarial Publications: Period Life Table 2005. Available at: www.ssa.gov/OACT/STATS/table4c6.html.ULM, E. R. 2009. The Effect of Policyholder Transfer Behavior on the Value of Guaranteed Minimum Death Benefits. Available at:

www.rmi.gsu.edu/Research/downloads/2009/Ulm RealTransfers.doc.

Discussions on this paper can be submitted until July 1, 2010. The authors reserve the right to reply to anydiscussion. Please see the Submission Guidelines for Authors on the inside back cover for instructions on thesubmission of discussions.

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