This article was downloaded by: [Laurentian University] On: 09 October 2013, At: 07:54 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 Valuing Equity-Indexed Annuities Serena Tiong A.S.A., Ph.D. a a GMAC-RFC , 8400 Normandale Lake Blvd., Suite 600 , Minneapolis, MN 55437 Published online: 04 Jan 2013. To cite this article: Serena Tiong A.S.A., Ph.D. (2000) Valuing Equity-Indexed Annuities, North American Actuarial Journal, 4:4, 149-163, DOI: 10.1080/10920277.2000.10595945 To link to this article: http://dx.doi.org/10.1080/10920277.2000.10595945 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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This article was downloaded by: [Laurentian University]On: 09 October 2013, At: 07:54Publisher: 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
Valuing Equity-Indexed AnnuitiesSerena Tiong A.S.A., Ph.D. aa GMAC-RFC , 8400 Normandale Lake Blvd., Suite 600 , Minneapolis, MN 55437Published online: 04 Jan 2013.
To cite this article: Serena Tiong A.S.A., Ph.D. (2000) Valuing Equity-Indexed Annuities, North American Actuarial Journal,4:4, 149-163, DOI: 10.1080/10920277.2000.10595945
To link to this article: http://dx.doi.org/10.1080/10920277.2000.10595945
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 of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe 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
Equity-indexed annuities have generated a great deal of interest and excitement among bothinsurers and their customers since they were first introduced to the marketplace in early 1995.Because of the embedded options in these products, the insurers are presented with somechallenging mathematical problems when it comes to the pricing and management of equity-indexed annuities. This paper explores the pricing aspect of three of the most common productdesigns: the point-to-point, the cliquet, and the lookback. Based on certain assumptions, we areable to present the pricing formulas in closed form for the three product designs. The method ofEsscher transforms is the fundamental tool for pricing such deferred annuities.
1. INTRODUCTION
Equity-indexed annuities (EIAs) have generated agreat deal of excitement and controversies sinceKeyport Life first launched “Key Index” in Febru-ary 1995. They are considered to be among themost innovative products to appear on the mar-ket in years. EIAs are, essentially, equity-linkeddeferred annuities whose returns are based on theperformance of an equity mutual fund or a familyof mutual funds or a stock index, typically of theStandard and Poor’s (S&P) 500 index.
An EIA has some basic features that make itsuch an attractive product to the customers.First, there is a minimum guarantee provision,usually at 3%, that eliminates the downside risk tothe customer. Second, the interest rates, oncecredited, are locked in, which means that theaccount value will never decrease as a result ofany downturns in the stock market. Third, be-cause the return is based on the performance of astock index, the customer can participate in thephenomenal growth of the stock market. Finally,EIAs are tax-deferred—the customers pay notaxes on the earnings until they make a with-drawal. From the insurers’ perspective, the min-imum guarantee provided is usually enough tomeet the nonforfeiture laws in various states, sothat the products can be sold as fixed annuities
and not as variable annuities; hence, they do notneed to be registered with the Securities Ex-change Commission (SEC) and the agents do notneed to be National Association of SecuritiesDealers (NASD)-registered to sell them. This al-lows the insurance companies to offer a form ofparticipation in equity markets without the com-plexities involved with registered products.
Product designs of EIAs can vary, depending onthe companies that sell them. In this paper, wediscuss the pricing and product design of some ofthe most commonly available EIA policies in themarket. The simplest one of all is a point-to-pointdesign, where the policy earns the realized returnon the index (or some other risky asset) over acertain period of time at a prescribed participa-tion rate, but with a minimum guarantee. Forexample, if the realized 10-year return on theS&P 500 is 100%, and we assume a participationrate of 90%, then the actual interest credited tothe policy will be 90% instead. Even when themarket performs poorly, the policy still earns areturn of at least the guaranteed rate.
The most favorable type of EIA product designseems to be the cliquet, which is French for ratchet.Under this cliquet design the interest rates are cred-ited annually, based on the higher of the indexreturn during the prevailing year, and a minimumguaranteed rate. This annual reset feature allowsthe resulting interest credits to be locked in, even ifthe index declines in the subsequent periods; there-fore, the value of the policy will never decrease.Another type of design that we will also touch on in
* Serena Tiong, A.S.A., Ph.D., is Vice President of Risk Analytics atGMAC-RFC, 8400 Normandale Lake Blvd., Suite 600, Minneapolis,MN 55437, e-mail: [email protected].
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this paper is the lookback, or the high-water-mark.This is also referred to as the “no-regret” type ofpolicy, which earn the highest return on the indexattained during the life of the policy. Cliquet andlookback EIAs are expensive by design; therefore,they usually offer a lower participation rate to com-pensate for the cost.
Many companies are still hesitant when it comesto offering EIAs, mainly because of the uncertaintyof reserving practices and regulations, the complex-ities in product designs and hedging, and the highcost of options. Due to falling interest rates anddividend yields and the rising option costs becauseof increased stock market volatility, the currentparticipation rates are considerably lower than theywere just a year ago. (See Mitchell and Slater 1996,Abkemeier 1999 and Horwitz 1999 for reports onthe EIA market.) Nevertheless, EIAs still present anappeal, especially to the low-risk-tolerant custom-ers because of their potential to earn equity-likereturns without subjecting the principal to unduerisk. The purpose of this paper is to provide someanswers and guidelines with respect to the pricingaspect of the EIAs, and help the readers gain betterunderstanding of the nature of these products.
The following assumptions are made for all thepricing formulas obtained in this paper. We as-sume that the market is complete; therefore,there exists a unique risk-neutral measure. Allthe underlying index or asset price processes fol-low the standard geometric Brownian motions.For simplicity, we also assume that the risk-freeforce of interest is constant, that there are notransaction costs, and that no lapse or death oc-curs before the policy matures. Throughout thepaper, the final payoff at maturity is assumed tobe based on an initial premium (or deposit) of $1at time 0, and all the pricing formulas are derivedusing the method of Esscher transforms.
2. ESSCHER TRANSFORMS
This section will serve as a quick summary ofGerber and Shiu (1994) on Esscher transforms.Descriptions on Esscher transforms can also befound in the books Bingham and Kiesel (1998)and Shiryaev (1999). Let S1(t) and S2(t) denotethe prices of two assets at time t, t $ 0, and let d1
and d2 be the respective constant nonnegativeinstantaneous dividend yield rates for asset 1 and2, such that the assets pay out dividends d1S1(t)dt
and d2S2(t)dt between time t and time t 1 dtrespectively. Let r be the constant risk-free forceof interest. We define
Yi~t! 5 ln@Si~t!/Si~0!#, i 5 1, 2
to be the rate of return on asset i over the timeinterval [0, t], and
Y~t! 5 ~Y1~t!, Y2~t!!9
to be the vector of returns. We assume that{Y(t)}t$0 is a stochastic process with independent(but not necessarily stationary) increments.
Let h 5 (h1, h2)9 be a nonzero real vector suchthat the joint moment generating function
M~h, t! 5 E@eh9Y~t!#
exists for all t $ 0. We define
M~z, t; h! 5 EFez9Y~t!eh9Y~t!
E@eh9Y~t!#G 5 E@ez9Y~t!; h#
to be the moment generating function of Y(t)under the Esscher transform with parameter vec-tor h. Note that
eh9Y~t!
E@eh9Y~t!#
is the Radon-Nikodym derivative used to definethe new probability measure.
A probability measure under which
$e2~r2di!tSi~t!%t$0, i 5 1, 2,
is called a martingale, or a risk-neutral probabilitymeasure. Gerber and Shiu (1994, 1996) showthat in the case where the assets are jointly amultidimensional geometric Brownian motion ora multidimensional geometric shifted compoundPoisson process, there exists a vector h*, calledthe risk-neutral Esscher parameter vector, suchthat
Es@e2~r2di!tSi~t!; h*# 5 e2(r2di)sSi~s!
for all s # t, where Es[ denotes the conditionalexpectation given all the information up untiltime s. If the market is complete, the risk-neutralmeasure and the corresponding h* are unique. Ifwe assume that Y(t) has a bivariate normal dis-tribution with mean mt 5 (m1t, m2t)9 and covari-ance matrix Vt 5 (sijt), i, j 5 1, 2, under thephysical measure, then h* is such that
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m 1 Vh* 5 r1 2 S d1 112 s11
d2 112 s22
D ,
where 1 5 (1, 1)9.
3. THE POINT-TO-POINT DESIGN
In the marketplace, some other names used to de-scribe this same product design are “European” and“end of term.” A variation of the point-to-point de-sign is the “average end” or “Asian end” design,where the average of a series of weekly or monthlyindices (typically in the year when the policy ma-tures) is used to establish the end point for deter-mining gains, thus diffusing the odds of having onebad down day. Here we shall discuss only the plainpoint-to-point design, where the index level at ma-turity is taken simply as the ending index. In allcases the starting index is the prevailing index levelwhen the policy is issued.
Let
S~t! 5 S~0!eY~t!, t $ 0,
be the value (or price) of an asset at time t thatpays out dividends dS(t)dt between time t andtime t 1 dt, where d is nonnegative. We can thinkof Y(t) as a random variable that denotes the(compounding) rate of return on the asset overthe time interval [0, t]. Let a be the participationrate, which, in practice, is almost always less thanor equal to 1. Suppose that at time T, T . 0, givenan initial premium of $1, we have a policy thatpays eaY(T), a . 0, or a fixed exercise price K(5elnK), K . 0, whichever is higher. Therefore, atmaturity, the policy earns a percentage of therealized return on the asset over T periods (theterm of the policy) which is aY(T), with the pro-vision of a minimum guaranteed rate of return, lnK. Hence, we can express the value of this policy,under the risk-neutral Esscher parameter h*, as
e2rTE@max~eaY~T!, K!; h*#. (3.1)
Because
max~eaY~T!, K! 5 eaY~T!I~aY~T! . ln K!
1 KI~aY~T! # ln K!,
where I[ is the indicator function (i.e., I(A) 5 1if A is true and 0 otherwise), we can further
decompose the expectation term on the right-hand side of Equation (3.1) into
E@eaY~T!I~aY~T! . ln K! 1 KI~aY~T! # ln K!; h*#
5 E@eaY~T!I~aY~T! . ln K!; h*#
1 KPSY~T! #ln K
a; h*D . (3.2)
By the method of Esscher transforms, the expec-tation term in Equation (3.2) (refer to Equation(A.2) in Appendix A and Proposition 2 in Appen-dix B for details) can be rewritten as
PSY~T! .ln K
a; h* 1 aDea~r2d!T1
12 a~a21!s2T,
(3.3)
where s is the volatility of the asset. If we followthe Black-Scholes assumptions that the price pro-cess {S(t)} is a geometric Brownian motion,which implies that Y(T) is normally distributedwith means
E~Y~T!; h*! 5 ~r 2 d 212 s2!T
and
E~Y~T!; h* 1 a! 5 ~r 2 d 212 s2 1 as2!T,
under Esscher transform for parameters h* andh* 1 a, respectively (proof in Appendix A), andvariance s2T (which is unaffected by the trans-form), then, together with Equation (3.3), we canexpress Equation (3.2), the value of this policy, as
Ppp 5 e@~a21!r2ad112 a~a21!s2#T
3 F1Sr 2 d 212
s2 1 as2DT 2ln K
a
sÎT2
1 e2rTKF1ln K
a2 Sr 2 d 2
12
s2DT
sÎT2 ,
(3.4)
where F[ denotes the cumulative distributionfunction of a standard normal random variable.
Let us now assume that the customer is guar-anteed a minimum maturity value, which is apercentage b, 0 , b # 1, of the original premiumcompounded at a minimum guaranteed rate of
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return g, g $ 0, for the duration of the policy. Wecan incorporate this feature into the policy bysetting the exercise price to be
K 5 begT, (3.5)
assuming an initial premium of $1. For most ofthe EIA policies sold by the insurance companies,b is set at around 90% and g at 3%.
Now we are going to make some observationsabout the function Ppp. In general, the value ofthe policy Ppp is an increasing function with re-spect to variables K, s and a, whereas it is de-creasing in d. As for r and T, the situations are notthat clear cut. Ppp can be increasing or/and de-creasing with respect to r and T, depending on thelevel of other variables, a in particular. For exam-ple, Figures 1, 2 and 3 clearly show the effect of aalone on Ppp. Within the context of our valuation,however, we can conclude that Ppp is decreasingin both r and T.
As we can see from Figure 1, when a is small,say less than 0.8, Ppp is almost a perfectly linearlydecreasing function of T. (If a is larger, say be-tween 0.8 and 1.2, Ppp increases in T for the firstfive to eight years, and then decreases linearlythereafter.) We can also show that for T less than10, Ppp is increasing linearly in a for a between0.5 and 1.1. Therefore, within the reasonablerange of these variables, we can approximate Ppp
by a linear regression model based on T or/and a;for example,
Ppp < a0 1 a1T 1 a2a 1 a3aT.
Assuming Equation (3.5) and an initial pre-mium of $1, we can solve Equation (3.4) implic-itly for a, such that the policy breaks even (inother words, Ppp 5 1). Table 1 lists the break evenparticipation rates for a under different scenar-ios. Using an a that is less (greater) than the breakeven, a is going to produce a value of Ppp that isless (greater) than one. Note that these participa-tion rates are considerably higher than those be-ing offered in the market under the same eco-nomic conditions.
Figure 3Plot of Ppp as a function of T given a 5 1.7(s 5 20%, r 5 5%, d 5 2%, K 5 0.9e0.03T)
Figure 1Plot of Ppp as a function of T given a 5 0.6(s 5 20%, r 5 5%, d 5 2%, K 5 0.9e0.03T)
Figure 2Plot of Ppp as a function of T given a 5 1.2(s 5 20%, r 5 5%, d 5 2%, K 5 0.9e0.03T)
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REMARK
Back in March 1987, a very similar type of prod-uct called market-index certificate of deposit(MICD) was introduced in the banking industryby Chase Manhattan Bank. The interest earnedon those certificates of deposit was contingentupon the performance of the S&P 500 index witha minimum guaranteed rate, just like the point-to-point EIAs analyzed in this section. These cer-tificates of deposit (CDs) were offered as a re-sponse to competition for funds during a long-running bull market, and they were considered areal breakthrough for significant profit potentialand an opportunity to draw business from themultibillion-dollar mutual fund industry. Thesame thing can be said about EIAs supplementingthe variable annuities business. The main differ-ence between MICDs and EIAs lies not in thestructural design of the products but in the regu-lations governing financial activities involvedwith those products. For more detailed descrip-tions on MICDs, interested readers are referred toChance and Broughton (1988), Chen and Kens-inger (1990), and King and Remolona (1987).
4. MAXIMUM OF TWO ASSETS
Suppose we have two assets where for i 5 1, 2,
Si~t! 5 Si~0!eYi~t!
is the value of asset i at time t, t $ 0, that pays outdividends diSi(t)dt between time t and t 1 dt,where di is nonnegative. Consider an EIA policythat credits the higher periodic return of thesetwo assets for n periods at a participation rate a,a . 0. The final payoff is made at time T. Note
that the periods can be of different lengths orduration. To simplify, we shall assume that eachperiod is of length m 5 T/n.
For i 5 1, 2 and j 5 1, 2, 3, . . . , n, let
Yij 5 Yi~ jm! 2 Yi~~ j 2 1!m!
denote the rate of return of asset i in period j. Foreach asset i, we assume that the periodic returnsare independent and identically distributed; how-ever, the returns of the two assets in the sameperiod can be correlated. For each period j, let
V 5 ~sik!
be a 2 by 2 matrix denoting the common covari-ance matrix of Yj 5 (Y1j, Y2j)9, and V is assumedto be nonsingular. Under the risk-neutral mea-sure, the value of this EIA policy (based on $1initial premium) at time 0 is
EFe2rT Pj51
n
eamax~Y1j,Y2j!; h*G , (4.1)
where h* 5 (h*1, h*2)9.We can rewrite Equation (4.1) as
e2rTEFPj51
n
~eaY1jI~Y1j . Y2j! 1 eaY2jI~Y1j # Y2j!!; h*G5 e2rT P
j51
n HE@~eaY1jI~Y1j . Y2j! 1 eaY2jI~Y1j # Y2j!!eh*9Yj#
E@eh*9Yj# J (4.2)
because the increments are assumed to be inde-pendent (see Lemma 2 in Appendix A). Note thatfor each j, j 5 1, 2, . . . , n,
Table 1Breakeven Participation Rates a for Point-to-Point EIA
This is the joint moment generating function of Yj
at z under the Esscher transform for parametervector h*. Similarly, and symmetrically, let 12 5(0 1)9. We have
E@eaY2jI~Y1j # Y2j!eh*9Yj#
E@eh*9Yj#(4.5)
5 P~Y1j # Y2j; h* 1 a12! Mj~a12; h*!. (4.6)
Thus, having Equation (4.4) and Equation(4.6), we can write Equation (4.2) as
e2rT Pj51
n
@P~Y1j . Y2j; h* 1 a11! Mj~a11; h*!
1 P~Y1j # Y2j; h* 1 a12! Mj~a12; h*!#. (4.7)
Since Yj’s are also assumed to be identically dis-tributed, then Equation (4.7) becomes
e2rT@P~Y1 . Y2; h* 1 a11! M~a11; h*!
1 P~Y1 # Y2; h* 1 a12! M~a12; h*!#n. (4.8)
Suppose each period is of length 1, and T 5 n.Define
Y 5 Y1~1! 2 Y2~1!
to be the excess periodic rate of return of asset 1over asset 2. Under the assumption that assetprices {S1(t)} and {S2(t)} are geometric Brownianmotions, Y is a normal random variable withmeans
E~Y; h* 1 a11!
5 ~a 212!s11 2 as12 1
12 s22 2 ~d1 2 d2!
; m1~a!
and
E~Y; h* 1 a12!
5 2 12 s11 1 as12 2 ~a 2
12!s22 2 ~d1 2 d2!
; m2~a!
under Esscher transform for parameter vectors h*1 a11 and h* 1 a12, respectively. The variance ofY, which is independent of the parameter vector,is
n2 ; s11 2 2s12 1 s22.
Therefore, we can write
P~Y . 0; h* 1 a11! 5 FSm1~a!
n D (4.9)
and
P~Y # 0; h* 1 a12! 5 FS2m2~a!
n D . (4.10)
Finally, with
M~a1i; h*! 5 exp~a~r 2 di 212 sii! 1
12 a2sii!,
i 5 1, 2,
Equation (4.8) or the value of this EIA policy attime 0 can be expressed as
e~a21!rnFe2ad1112 a~a21!s11FSm1~a!
n D1 e2ad21
12 a~a21!s22FS2m2~a!
n DGn
. (4.11)
Now assume that a 5 1. Observe that if d1 'd2 5 d, then d1 2 d2 ' 0 and
m1~1! < 2m2~1! < n2/ 2,
hence, Equation (4.11) becomes (approximately)
@2e2dF~n/ 2!#n. (4.12)
Because n is strictly positive, which implies that
F~n/ 2! .12 ,
if d is small enough such that
2e2dF~n/ 2! . 1
or
0 # d , ln@2F~n/ 2!# <n
Î2p, (4.13)
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then the present value or price of this EIA isgreater than 1 and therefore increasing with n. Aninterpretation of this phenomenon is that, whenthe dividend yields are small enough, such thatthe inequality above holds, the dividends that thecustomer forgoes by holding this EIA policy in-stead of the actual assets will be smaller than thecontractual return on this policy, which is thecompounded higher annual return of the two as-sets. Therefore the policy will have a presentvalue that is greater than 1. Conversely, if
d .n
Î2p, (4.14)
then the price is less than 1 and decreasing withn. In this case, when the dividend yields are large,by holding the policy, the customer forgoes morethan he can earn if he holds the actual assets.Obviously the breakeven point occurs when
d <n
Î2p, (4.15)
and the price of the EIA will be approximately 1,which is equal to the initial premium. In order tobreak even, when Equation (4.13) is true, we willneed an a that is less than 1, and conversely a hasto be greater than 1 to break even when Equation(4.14) is true instead.
In general, for any given set of dividend yields,(d1, d2), where di is nonnegative for i 5 1, 2, wecan find a participation rate a (implicitly) suchthat the EIA policy breaks even, that is, Equation(4.11) equals 1 or
e~a21!rHe2ad1112 a~a21!s11FSm1~a!
n D1 e2ad21
12 a~a21!s22FS2m2~a!
n DJ 5 1.
5. THE CLIQUET DESIGN
The following excerpt from Lo (1997) can serve asthe motivation for this particular product design:
If, in January of 1926, an investor put $1 intoone-month U.S. Treasury bills—one of the “saf-est” assets in the world—and continued reinvest-ing the proceeds in Treasury bills month bymonth until December 1994, the $1 investmentwould have grown to $12. If, on the other hand,
the investor had put $1 into the stock market, e.g.,the S&P 500, and continued reinvesting the pro-ceeds in the stock market month by month overthis same 68-year period, the $1 investmentwould have grown to $811, a considerably largesum. Now suppose that each month, an investorcould tell in advance which of these two invest-ments would yield a higher return for that month,and took advantage of this information by switch-ing the running total of his initial $1 investmentinto the higher-yielding asset, month by month.What would a $1 investment in such a “perfectforesight” investment strategy yield by December1994?
The startling answer—$1,251,684,443 (yes,over one billion dollars, this is no typographicalerror)—often comes as a shock to even the mostseasoned professional investment manager. Ofcourse, few investors have perfect foresight, butthis extreme example suggests that even a modestability to forecast financial asset returns may behandsomely rewarded: It does not take a largefraction of $1,251,684,443 to beat $811!
Now let us consider an EIA with a cliquet de-sign, of which the basic concept is closely relatedto the above illustration by Lo. We assume that, atthe end of every period, the annuity will earn theperiodic return on an asset {S(t)} with a partici-pation rate a, a $ 0, or the minimum guaranteedreturn of g, g $ 0, whichever is higher. Once theinterest is credited, the earnings are locked in andwill never decrease, regardless of the future per-formance of the market. We also assume that theasset pays out dividends dS(t)dt between time tand t 1 dt, where d is nonnegative, and the finalpayoff is made at the end of n periods. Thus, thevalue of this EIA policy, at time 0, is
EFe2rn Pj51
n
max~eg, eaYj!; h*G , (5.1)
where
Yj 5 Y~ j! 2 Y~ j 2 1!
denotes the rate of return on the asset in period j,j 5 1, 2, . . . , n. We assume that {Yj} is a sequenceof independent and identically distributed ran-dom variables. Note that the cliquet design is aspecial case of the previous two-asset EIA poli-cies, where the second asset is nonrandom andearns a fixed rate of return. Let us put
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Y1j 5 Yj,
Y2j 5 g/a,
d1 5 d,
d2 5 0,
h* 5 ~h* 0!9,
then Equation (4.3) and therefore Equation (4.4)become
PSYj .g
a; h* 1 aDea~r2d!1
12 a~a21!s2 (5.2)
and Equation (4.5) can now be written as
egEFISYj #g
aDeh*YjGE@eh*Yj#
5 egPSYj #g
a; h*D . (5.3)
Substituting Equation (5.2) and Equation (5.3)into Equation (4.2), the value of this cliquet pol-icy can now be expressed as
e2rn Pj51
n Fea~r2d!112 a~a21!s2PSYj .
g
a; h* 1 aD
1 egPSYj #g
a; h*DG
5 Pj51
n Fe2~12a!r2ad112 a~a21!s2PSYj .
g
a; h* 1 aD
1 e2~r2g!PSYj #g
a; h*D . (5.4)
If we also assume that the process {S(t)} is ageometric Brownian motion, then Yj’s are identi-cally normally distributed with variance s2, andfor each j, Yj has means r 2 d 2 1
2 s2 and r 2 d 1(a 2 1
2)s2 under Esscher transform for parameters
h* and h* 1 a respectively. Equation (5.4) nowbecomes
Pc 5 @e2~12a!r2ad112 a~a21!s2
3 FS2g/a 1 r 2 d 2 s2/ 2 1 as2
s D
1 e2~r2g!FSg/a 2 r 1 d 1 s2/ 2s Dn
.
(5.5)
5.1 The Cliquet With Cap DesignSuppose now that we place a fixed upper limit, orcap, on the periodic return. Let c be the capwhere c . g. This will be the maximum rate ofinterest that the policy can earn in each period.Based on all the same assumptions as stated inthe previous section, the value of the EIA policynow becomes
EFe2rn Pj51
n
max~eg, min~ec, eaYj!!; h*G . (5.6)
To simplify writing, define
a ~ b 5 max~a, b!
and
a ` b 5 min~a, b!;
then we can express Equation (5.6) as
EFe2rn Pj51
n
eg~~aYj`c!; h*G5 e2rn P
j51
n
E@eg~~aYj`c!; h*# (5.7)
by the independence of the random variables {Yj}.Note that since we can write
eg~~aYj`c! 5 egI~aYj # g! 1 eaYjI~ g , aYj # c!
1 ecI~aYj . c!,
we can express
E@eg~~aYj`c!; h*# 5 egPSYj #g
a; h*D
1 ea~r2d!112 a~a21!s2PSg
a, Yj #
c
a; h* 1 aD
1 ecPSYj .c
a; h*D . (5.8)
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Thus, substituting Equation (5.8) into Equation(5.7), the price of this EIA with cap c is
Pj51
n Fe2~r2g!PSYj #g
a; h*D
1 e2~12a!r2ad112 a~a21!s2PSg
a, Yj #
c
a; h* 1 aD
1 ec2rPSYj .c
a; h*DG . (5.9)
Since Yj’s are identically and normally distributedwith variance s2, then Equation (5.9) becomes
Pcc 5 Fe2~r2g!FSg/a 2 r 1 d 1 s2/ 2s D
1 ec2rFSr 2 d 2 s2/ 2 2 c/as D
1 e2~12a!r2ad112 a~a21!s2
3 HFSc/a 2 r 1 d 1 s2/ 2 2 as2
s D2 FSg/a 2 r 1 d 1 s2/ 2 2 as2
s DJGn
.
(5.10)
Note that Equation (5.5) is a special case of Equa-tion (5.10) when c 5 `.
Let us define
D 5 ~Pcc!1/n 2 1,
then we can express Equation (5.10) as
Pcc 5 ~1 1 D!n.
Obviously, Pcc is increasing (decreasing) in n if Dis positive (negative). Within the realistic range ofa, c and other variables, D is usually small enoughthat Pcc can be well approximated by a linearfunction in n; hence,
Pcc < 1 1 nD.
Observe that, since c . g, we always have
eg # eg~~aYj`c! # ec,
and, therefore, in Equation (5.7), e(c2r)n ande(g2r)n become an upper and lower bound for thevalue of this policy at time 0. Note that e(c2r)n isless than 1 (the initial premium) if c , r.
For any finite c, we can see from Figure 4 thatas a function of a, Pcc first increases, then flattensout and eventually converges to
Fe2~r2g!FS2r 1 d 1 s2/ 2s D
1 ec2rFSr 2 d 2 s2/ 2s DGn
as a approaches `. This is due to the fact thatregardless of the level of participation, the policyearns no more than a rate of c in each period. Alsonote that the larger c is, the steeper the slope of thefunction and therefore the slower the convergence.
Table 2 lists the breakeven participation ratesunder different scenarios, (as such that Pcc 5 1).These breakeven participation rates are indepen-dent of the time to maturity T, and they areconsiderably lower than those of the point-to-point design (see Table 1); hence, the higherprices for cliquet EIAs. Notice how sensitivelythese participation rates respond to the risk-freeforce of interest r when c is low. One can see thatall the breakeven participation rates listed in Ta-ble 2 are decreasing in s, except for when r 5 6%and c 5 10%. In this case, we actually have ahigher breakeven participation rate when s 530% than when s 5 20%. A way to see this is that,when the asset becomes more volatile, the rangeof possible rates of return widens accordingly.
Figure 4Plots of Pcc (n 5 1) as a function of a with
different values of c (s 5 20%, r 5 5%,d 5 2%, g 5 3%)
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While the returns are capped by c, higher realizedreturns cannot be captured; it is then more likelyfor the policy to earn a lower return. So if r is largeenough (that is, the discounting is steep enough),we get a lower price for the policy, which, in turn,results in a higher breakeven participation rate.
5.2 The Cliquet With Spread DesignA great deal of interest in the “spread” type of EIAproducts, especially of the cliquet design, wasshown in 1998. This type of product typicallyearns a 100% participation in growth of the S&P500 in excess of a predetermined hurdle rate,which can be termed as a spread or a “fee,” alongwith some minimum guarantee. When comparedto the cliquet products with cap, the spread EIAsare obviously much more attractive in the bullmarkets (that is, when the actual S&P 500 returnsare high), because they offer the potential to earnunlimited upside benefits.
Let s be the spread or the hurdle level, s $ 0,and a the participation rate. We assume that,given a minimum guaranteed periodic return of g,for each period, the policy will earn a return ofmax(g, a(Yj 2 s)). In other words, the creditedreturn is a(Yj 2 s) if a(Yj 2 s) . g, (that is, ifthe actual return on the index is greater than s 1 g
a)
and the minimum guarantee g if otherwise. Thus,the value of this EIA policy is
EFe2rn Pj51
n
max~eg, ea~Yj2s!!; h*G5 e2rn P
j51
n
E@max~eg, ea~Yj2s!!; h*#. (5.11)
Note that because
max~eg, ea~Yj2s!! 5 egI~a~Yj 2 s! # g!
1 ea~Yj2s!I~a~Yj 2 s! . g!,
Equation (5.11) becomes
Pj51
n Fe2~r2g!PSYj # s 1g
a; h*D
1 e2~12a!r2a~s1d!112 a~a21!s2
3 PSYj . s 1c
a; h* 1 aDG . (5.12)
By the assumption that Yj’s are identically andnormally distributed with variance s2, then Equa-tion (5.12) becomes
Pcs 5 Fe2(r2g)FSs 1 g/a 2 r 1 d 1 s2/ 2s D
1 e2~12a!r2a~s1d!112 a~a21!s2
3 FSr 2 d 2 s2/ 2 1 as2 2 s 2 g/as DGn
.
(5.13)
Note that Equation (5.5) is a special case of Equa-tion (5.13) when s 5 0.
One interesting observation that we can makefrom Equation (5.13) is that, since s always ap-pears along side d, we can conclude, with thefollowing result that
Pcs~d! 5 Pc~d 1 s!,
that is, the price of a cliquet with spread s equals
Table 2Breakeven Participation Rates a for Cliquet EIA
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the price of the “unrestrained” cliquet (as dis-cussed in section 5) with its dividend yield delevated by s. In other words, it is as if the insureris making a higher dividend yield assumption inthis case. The customer’s point of view would bethat he or she is always earning a higher return onthe cliquet without spread, although, to the in-surer, the two are equivalent in terms of pricingbecause of the difference in the dividend yieldassumption.
6. THE CONTINUOUS LOOKBACK DESIGN
Another popular design among the EIA productsis the lookback or high-water-mark method. Theidea is that, at maturity, the interest earned onthe policy will be based on the growth rate of thehighest index value attained during the life of thepolicy over the index value at the start of theterm, which we assume to be one in our calcula-tions. In practice, the method usually looks at theindex level at each policy anniversary, and thehighest of these is then taken and figured as theindex level on the maturity date. Here we con-sider the continuous lookback case, where wehave a nice closed-form formula for the value ofthis type of policy.
Let
M~T! 5 max0#t#T
Y~t!
be the maximum rate of return on the indexattained over the time interval [0, T]. We assumethat the payoff of the policy at time T is eaM(T),0 , a # 1, or a fixed minimum guaranteedamount, K, whichever is larger. In this case, attime 0, the value of this policy is
E@e2rT max~eaM~T!, K!; h*#. (6.1)
Under the assumption that the index process is ageometric Brownian motion, Equation (6.1) canbe expressed as (see Appendix B for details)
Plb 5 e2rT1h~a!F~d1~a! 1 asÎT!
1 e2rT1m~a!F~d2~a! 1 g~a!sÎT!
2 e2rT2~r 2 d! 2 s2
g~a!s2 @2eg~a!k~a!F~d2~a!!
1 em~a!F~d2~a! 1 g~a!sÎT!#
1 e2rTK@F~2d1~a!!
2 e$@2~r2d!2s2#k~a!%
s2 F~d2~a!!# (6.2)
where
k~a! 5ln K
a,
g~a! 5 a 12~r 2 d!
s2 2 1,
h~a! 5 a~r 2 d!T 112 a~a 2 1!s2T,
m~a! 5 2g~a!~r 2 d!T 112 g~a!~ g~a! 2 1!s2T,
d1~a! 5~r 2 d 2
12 s2!T 2 k~a!
sÎT,
d2~a! 52~r 2 d 2
12 s2!T 2 k~a!
sÎT.
Note that we can rewrite Plb as the sum of twocomponents: the value of a European continuouslookback call option Clb and the value of the fixedexercise price K at time 0; hence,
Plb 5 e2rTK 1 Clb
where
Clb 5 e2rTE@max~eaM~T! 2 K, 0!; h*#.
REMARK
When a 5 1, Clb becomes the price of a standardfixed strike lookback call option. In this case, weget the exact same formula for Clb 5 Plb 2 e2rTKas in Conze and Viawanathan (1991) when d 5 0,and Haug [1998, formula (2.33)], Wilmott (1998,p. 232) and Zhang [1998, formulas (12.13) and(12.14)] when d $ 0.
7. CONCLUSIONS AND FUTURE RESEARCH
Due to the current volatile economic environ-ment that leads to lower participation rates, salesof the EIA products have shown signs of slowingdown. However, many still believe that EIAs arehere to stay, as they address a genuine need for alarge segment of the population—those with a lowrisk tolerance who want a conservative alterna-tive for part of their retirement planning or sav-
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ings. Market conditions will determine whenthese EIA products will move again to the fore-front; the bottom line is, EIAs offer an excellenttool for helping people manage their asset alloca-tion and risk profile and the opportunity to par-ticipate in the equity market with protection forthe principal. With all these possibilities, EIAsdeserve a position in both the insurers’ and cus-tomers’ portfolio.
This paper presents the pricing formulas for threeof the most common EIA product designs currentlyavailable in the marketplace, namely the point-to-point, the cliquet, and the lookback. Based on ourassumptions of constant risk-free force of interest,no lapse, no death, no transaction costs, and a per-fect market, we are able to obtain those results inclosed form. More research is needed to incorporatemore realistic assumptions into the pricing models.Some areas of interest are stochastic interest rates,stochastic volatility, lapse rates, and death benefits.As product designs get more complicated than wehave seen in the marketplace, more advancedmathematical tools are needed to ensure correctpricing. In addition, studies on the reserving, hedg-ing and investment strategies needed to appropri-ately manage these products are also of great im-portance.
ACKNOWLEDGMENTS
I gratefully acknowledge the Ph.D. grant from theCasualty Actuarial Society and the Society of Ac-tuaries. I would also like to thank Professors Shel-don Lin and Elias Shiu for their guidance and Mr.Jack Marrion of the Advantage Group, St. Louis,for providing me with the very informative IndexAnnuities Report.
ReferencesABKEMEIER, N. 1999. “What Lies Ahead for EIAs,” National Un-
derwriter (Life and Health/Financial Services Edition) 103(10):7.
BINGHAM, N. H., AND KIESEL, R. 1998. Risk-Neutral Valuation:Pricing and Hedging of Financial Derivatives. London:Springer.
BLACK, F., AND SCHOLES, M. 1973. “The Pricing of Options andCorporate Liabilities,” Journal of Political Economy 81:637–54.
CHANCE, D. M., AND BROUGHTON, J. B. 1988. “Market Index DepositoryLiabilities: Analysis, Interpretation, and Performance,” Jour-nal of Financial Services Research 1:335–52.
CHEN, A. H., AND KENSINGER, J. W. 1990. “An Analysis of Market-
Index Certificates of Deposit,” Journal of Financial Ser-vices Research 4:93–110.
CONZE, A., AND VISWANATHAN 1991. “Path Dependent Options: TheCase of Lookback Options,” Journal of Finance 46:1893–1907.
COX, J. C., AND RUBINSTEIN, M. 1985. Option Markets. EnglewoodCliffs, N.J.: Prentice Hall.
GERBER, H. U., AND SHIU, E. S. W. 1994. “Option Pricing byEsscher Transforms,” Transactions of the Society of Actu-aries 46:99–140.
GERBER, H. U., AND SHIU, E. S. W. 1996. “Actuarial Bridges toDynamic Hedging and Option Pricing,” Insurance: Mathe-matics and Economics 18:183–218.
GOLDMAN, M. B., SOSIN, H. B., AND GATTO, M. A. 1979. “PathDependent Options: Buy at the Low, Sell at the High,”Journal of Finance 34:1111–27.
HARRISON, J. M. 1985. Brownian Motion and Stochastic FlowSystems. New York: Wiley.
HAUG, E. P. 1998. The Complete Guide to Option Pricing For-mulas. New York: McGraw-Hill.
HORWITZ, E. J. 1999. “EIAs Still Provide Competitive Potential,”National Underwriter (Life and Health/Financial ServicesEdition) 103(2):13.
HULL, J. C. 1997. Options, Futures, and Other Derivative Secu-rities, 3rd ed. Englewood Cliffs, N.J.: Prentice Hall.
KARR, A. F. 1993. Probability. New York: Springer-Verlag,KING, S. R., AND REMOLONA, E. M. 1987. “The Pricing and Hedging
of Market Index Deposits,” Federal Reserve Bank NewYork Quarterly Review 11 (Summer):9–20.
LO, A. W. 1997. “A Non-Random Walk Down Wall Street,” in TheLegacy of Nobert Wiener: A Centennial Symposium, Prov-idence, R.I.: American Mathematical Society, pp. 49–83.
MITCHELL, G. T., AND SLATER, J., JR. 1996. “Equity-Indexed Annu-ities-New Territory on the Efficient Frontier,” Society ofActuaries Study Note Number 441:99–96.
PANJER, H. H., ed., et al. 1998. Financial Economics: With Ap-plications to Investments, Insurance, and Pensions.Schaumburg, Ill.: The Actuarial Foundation.
SHIRYAEV, A. N. 1999. Essentials of Stochastic Finance: Facts,Models, Theory. Singapore: World Scientific.
TIONG, S. 2000. “Some Pricing Formulas for Equity-Indexed An-nuities,” Actuarial Research Clearing House 2000.1, 353.
WILMOTT, P. 1998. Derivatives: The Theory and Practice of Fi-nancial Engineering. New York: Wiley.
ZHANG, P. G. 1998. Exotic Options: A Guide to Second Genera-tion Options, 2nd ed. Singapore: World Scientific.
APPENDIX A
SOME BASICS
Suppose Y is a normal random variable with meanm and variance s2 under the physical measurewith probability space (V, ^, P). For any realnumber z, the moment generating function of Yunder Esscher transform with respect to param-eter h is
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MY~ z; h! ; E@ezY; h#
5E@ezYehY#
E@ehY#
5exp$~h 1 z!m 1
12 ~h 1 z!2s2%
exp$hm 112 h2s2%
5 expH ~m 1 hs2! z 112
s2z2J , (A.1)
which is the moment generating function for anormal random variable with mean m 1 hs2 andvariance s2.
Let A be an event; that is, A [ ^, and a anarbitrary real number. Then, under the Esschertransform for parameter h, we can write
E@eaYI~ A!; h# 5E@eaYI~ A!ehY#
E@ehY#
5E@I~ A!e~h1a!Y#
E@e~h1a!Y#
E@e~h1a!Y#
E@ehY#
5 P~ A; h 1 a! MY~a; h!. (A.2)
Let Y(T) be the rate of return on an asset overtime interval [0, T]; that is, Y(T) is normal withmean mT and variance s2T under the physicalmeasure. By the Fundamental Theorem of AssetPricing, the risk-neutral Esscher parameter h* isa constant such that
MY~T!~1; h*! 5 e~r2d!T,
or
m 1 h*s2 5 r 2 d 212 s2.
Then Equation (A.1) implies that
MY~T!~ z; h*! 5 exp$~r 2 d 212 s2!Tz 1
12 s2Tz2%
and
MY~T!~ z; h* 1 a!
5 exp$~r 2 d 212 s2 1 as2!Tz 1
12 s2Tz2%;
in other words, Y(T) is normally distributed withmeans (r 2 d 2 1
2 s2)T and (r 2 d 2 12 s2 1 as2)T
under Esscher transform for parameters h* andh* 1 a respectively, and common variance s2T.
LEMMA 1
Let U and V be two random variables. Then U andV are independent if and only if, for any positivefunctions f and g,
E@ f~U! g~V!# 5 E@ f~U!#E@ g~V!#.
PROOF
(f) See Corollary 4.30 in Karr (1993).(d) Simply take f and g to be indicator func-
tions, and the result follows from the definition ofindependence.
LEMMA 2
Let U and V be two underlying random variables.Suppose they are independent under the originalmeasure. Then, U and V remain independent un-der the Esscher measure for each parameter h.
PROOF
We want to show that
E@ f~U! g~V!; h# 5 E@ f~U!; h#E@ g~V!; h#
for all positive functions f and g. Then, by Lemma1, we can conclude that U and V are independentunder the new measure corresponding to Esscherparameter h.
Suppose that U and V are independent underthe original measure. By the definition of Esschertransforms, for any arbitrary positive functions fand g,
E@ f~U! g~V!; h# 5E@ f~U! g~V!eh~U1V!#
E@eh~U1V!#
5E@ f~U!ehU#
E@ehU#zE@ g~V!ehV#
E@ehV#by Lemma 1
5 E@ f~U!; h#E@ g~V!; h#.
The proof is completed.Note that Equation (5.5) and Equation (5.10)
are derived assuming that Yj’s are identically dis-tributed with the same distribution as Y(1).
APPENDIX B
THE PROOF OF (6.2)Let {Y(t)}t$0 be a Brownian motion with drift mand volatility s. Then for any t $ 0, Y(t) is anormal random variable with mean mt and vari-ance s2t. Define
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M~t! 5 max0#s#t
Y~s!.
The cumulative distribution and probabilitydensity functions of M(t) are given in the follow-ing proposition.
Proposition 1
For t . 0, y $ 0,
P~M~t! # y! 5 FSy 2 mt
sÎt D 2 e2my/s2FS2y 2 mt
sÎt D ,
and
fM~t!~ y! 51
sÎtfSy 2 mt
sÎt D1 e2my/s2 1
sÎtfS2y 2 mt
sÎt D2
2m
s2 e2my/s2FS2y 2 mt
sÎt D ,
where F[ and f[ denote the cumulative distri-bution and probability density function of a stan-dard normal random variable, respectively.
PROOF
The cumulative distribution function of M(t) isproved using the reflection principle; see Harri-son (1985) for details. The probability densityfunction, fM(t)(y), is obtained by differentiatingP(M(t) # y) with respect to y.
Proposition 2
Suppose that a random variable X is normallydistributed with mean m and variance s2. Thenfor any real numbers a and z,
E@ezXI$X . a%# 5 FS ~m 1 s2z! 2 a
s Demz1s2z2/ 2.
PROOF
Apply Equation (A.2) with Y 5 X, a 5 z, h 5 0 andA 5 {X . a}. It is well-known that MX(z) 5exp(mz 1 1
2 s2z2); and by Equation (A.1), X isnormal with mean m 1 s2z and unchanged vari-ance s2 under the Esscher transform for param-eter z; therefore
P~X . a;z! 5 FS ~m 1 s2z! 2 a
s D .
REMARK
Under the risk neutral transform h*,
m 5 r 2 d 212 s2
in Propositions 1 and 2.Because
max~eaM~T!, K! 5 eaM~T!I~aM~T! . ln K!
1 KI~aM~T! # ln K!,
the value of the continuous lookback EIA policyat time 0 can be expressed as
Plb 5 e2rTE@eaM~T!I~aM~T! . ln K!; h*#
1 e2rTKP~aM~T! # ln K; h*!. (B.1)
Let
k~a! 5ln K
a.
By Proposition 1, the probability term in Equa-tion (B.1) becomes
P~M~T! # k~a!; h*!
5 FSk~a! 2 ~r 2 d 2 s2/ 2!T
sÎT D2 expS @2~r 2 d! 2 s2#k~a!
s2 D3 FS2k~a! 2 ~r 2 d 2 s2/ 2!T
sÎT D . (B.2)
The expectation term in (B.1) is
E@eaM~T!I~aM~T! . ln K!; h*#
5 Ek~a!
`
eayfM~T!~ y!d y 5 E@eaX1I$X1 . k~a!%#
1 E@eg~a! X2I$X2 . k~a!%# 2 S2~r 2 d!
s2 2 1D3 E
k~a!
`
eg~a! yFS2y 2 ~r 2 d 2 s2/ 2!T
sÎT Dd y,
(B.3)where
g~a! 5 a 12~r 2 d!
s2 2 1,
and X1 and X2 are normal random variables with
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means (r 2 d 2 12 s2)T and 2(r 2 d 2 1
2 s2)T,respectively, and a common variance s2T. Then,by Proposition 2, the first two expectations inEquation (B.3) can be simplified as
eh~a!FS ~r 2 d 2 s2/ 2 1 as2!T 2 k~a!
sÎT D (B.4)
and
em~a!FS ~2r 1 d 1 s2/ 2 1 s2g~a!!T 2 k~a!
sÎT D ,
(B.5)
where
h~a! 5 a~r 2 d!T 112 a~a 2 1!s2T
and
m~a! 5 2g~a!~r 2 d!T 112 g~a!~ g~a! 2 1!s2T.
Using integration by parts and Proposition 2,we can rewrite the integration term in Equation(B.3) as
1g~a! E
k~a!
`
FS2y 2 ~r 2 d 2 s2/ 2!T
sÎT Dd~eg~a! y!
51
g~a! Feg~a! yFS2y 2 ~r 2 d 2 s2/ 2!T
sÎT DU y5k~a!
y5`
1 Ek~a!
`
eg~a! y1
sÎTfS2y 2 ~r 2 d 2 s2/ 2!T
sÎT Dd yG5
1g~a! F2eg~a!k~a!FS2k~a! 2 ~r 2 d 2 s2/ 2!T
sÎT D1 em~a!FS ~2r 1 d 1 s2/ 2 1 s2g~a!!T 2 k~a!
sÎT DG .
(B.6)
Formula (6.2) then follows.
DISCUSSIONS
G. THOMAS MITCHELL*As an actuarial practitioner in various areas in-volving promises of equity returns with guaran-tees, I thank the author for the very useful ana-lytic formulas she presents in this paper.
The main formulas are easily implemented in aspreadsheet format. They will be especially usefulfor the following purposes:
1. Gaining a feel and insight into option pricingand cost of benefits
2. Fine-tuning pricing and hedging as the optionsmarket fluctuates
3. Avoiding the complications of Monte Carlopricing in some cases.
4. Computing reserves.
The values derived in the paper don’t represent afull pricing model for equity-indexed annuities.Considerations of mortality, lapses, dynamics ofpolicyholder behavior under varying market con-ditions, yield curve movement, expenses, taxes,and reserving all add complexities. The end resultis that a Monte Carlo model is generally needed todo the whole job.
Further insight can be gained by studying thesensitivity of the value of the policy to the variousparameters used in the paper, particularly bycomparing the different participation structures.One important issue is the relative sensitivity tovolatility, in light of the major instability in vola-tility recently.
I implemented the main formulae in a spread-sheet and calculated the approximate partial de-rivatives of the value of the policy (Table 1). Notethat the point-to-point has a typical 10% load andthe cliquet examples do not.
Perhaps the most important observation is thesensitivity to volatility. Point-to-point and un-capped cliquet designs are much more sensitiveto volatility than a capped cliquet. The uncappedcliquet is the most sensitive. If the market fluc-tuates wildly over the seven-year term, the clientbenefits from the up years, even if overall perfor-
* G. Thomas Mitchell, F.S.A., M.A.A.A., is President of Aurora Con-sulting Inc., 8630 Delmar Boulevard, Suite 200, St. Louis, MO 63124-2208, e-mail: [email protected].
