The application of an accurate approximation in the risk management of investment guarantees in life insurance

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1 The applicatio of a accurate approximatio i the risk maagemet of ivestmet guaratees i life isurace Kobus Bekker a, Ja Dhaee b a ABSA Life, 21 Kruis Street, Johaesburg, South Africa. b K.U.Leuve, 69 Naamsestraat, B-3000 Leuve, Belgium. Abstract Ivestmet guaratees i life isurace busiess have geerated a lot of research recetly due to the earlier mispricig of such products. These guaratees geerally take the form of exotic optios ad are therefore di cult to price aalytically, eve i a simpli ed settig. A possible solutio to the risk maagemet problem of ivestmet guaratees cotiget o death ad survival is proposed through the use of a coditioal lower boud approximatio of the correspodig embedded optio value. The derivatio of the coditioal lower boud approximatio is outlied i the case of regular premiums with asset-based charges ad the implemetatio is illustrated i a Black-Scholes-Merto settig. The derived coditioal lower boud approximatio also facilitates verifyig ecoomic sceario geerator based pricig ad valuatio, as well as sesitivity measures for hedgig solutios. Keywords: Ivestmet Guaratees, Comootoicity, Life Isurace October 5, 2010

2 The applicatio of a accurate approximatio i the risk maagemet of ivestmet guaratees i life isurace October 5, 2010 Abstract Ivestmet guaratees i life isurace busiess have geerated a lot of research recetly due to the earlier mispricig of such products. These guaratees geerally take the form of exotic optios ad are therefore di cult to price aalytically, eve i a simpli ed settig. A possible solutio to the risk maagemet problem of ivestmet guaratees cotiget o death ad survival is proposed through the use of a coditioal lower boud approximatio of the correspodig embedded optio value. The derivatio of the coditioal lower boud approximatio is outlied i the case of regular premiums with assetbased charges ad the implemetatio is illustrated i a Black-Scholes- Merto settig. The derived coditioal lower boud approximatio also facilitates verifyig ecoomic sceario geerator based pricig ad valuatio, as well as sesitivity measures for hedgig solutios. 1

3 1 Itroductio Life isurers have traditioally cocered themselves with maily mortality risk, which is a diversi able risk if the isurer is able to aggregate a large umber of idepedet isured lives. The risks iheret to ivestmet guaratees, however, are largely depedet ad require a approach to fair pricig of these guaratees ad a hedgig programme to e ectively trasfer ivestmet risk to third parties or market. Accordig to Hardy (2003), the three mai aspects of risk maagemet relates to the price, the amout of capital to hold ad how to ivest the capital. Iitial attempts at pricig ivestmet guaratees were maily of a statistical real-world approach. Sigi cat improvemets i the eld of ace with the papers by Black ad Scholes (1973) ad Merto (1973) led to attempts to combie the elds of actuarial sciece ad ace. Of the rst substatial attempts were the publicatios by Boyle ad Schwartz (1977), Brea ad Schwartz (1977) ad Brea ad Schwartz (1979). The last metioed authors ot oly cosidered fair pricig of both sigle ad recurrig premium structures i a Black-Scholes-Merto framework, but also looked at a possible delta hedgig strategy ad the sesitivities of the hedgig strategy to model parameters. The martigale approach to risk-eutral pricig by Harriso ad Kreps (1979) ad Harriso ad Pliska (1981) further facilitated the adoptio of moder ace techiques by the actuarial professio. The stadard Black- Scholes-Merto framework uder the martigale approach was applied to miimum guaratees at death ad maturity by, amog others, Delbae (1986) 2

4 ad Aase ad Persso (1994). The complexity i bee ts as well as i the assumptios uderlyig the approaches cosidered i literature have grow substatially. Baciello ad Ortu (1993a) cosidered edogeous miimium guaratees, i.e. miimum guaratees that deped fuctioally o the premium of the policies. Baciello ad Ortu (1993b), Nielse ad Sadma (1995) ad Nielse ad Sadma (1996) exteded existig results to iclude stochastic iterest rate risk. Boyle ad Hardy (1997) cosidered a Value at Risk (VaR) methodology ad a dyamic replicatig portfolio approach. Møller (1998) cosidered risk-miimisig strategies set i a Black- Scholes-Merto framework ad Møller (2001) proposed a more pragmatic risk-miimisig strategy i a discrete Cox-Ross-Rubistei framework. The policyholder cotributios are usually solved for uder fair value priciples. We assume that the policyholder cotributios are exogeously give, which is largely the case i practice. The paymet of regular premiums results i the payo beig depedet o the uderlyig asset price throughout the duratio of the cotract ad leads to a aalogy with pathdepedet Asia optios. Upper ad lower bouds i terms of double itegrals for approximately valuig Asia optios have bee developed by Rogers ad Shi (1995) ad Thompso (1999). The applicatio of stochastic bouds to acial products i actuarial sciece was itroduced maily by Simo et al. (2000), Dhaee et al. (2002a,b), Nielse ad Sadma (2003) ad Shrager ad Pelsser (2004). The last metioed authors use a chage of umeraire techique to derive a geeral pricig formula i the case of stochastic iterest rates i the logormal case for rate of retur guaratees i 3

5 regular premium busiess. Hürlima (2008) cosiders GMDB ad GMAB guaratees i regular premium uit-liked (UL) busiess i the logormal case with a two-factor fud di usio process, cosistig of a oe-factor fud price ad a oe-factor stochastic iterest rate model with determiistic bod price volatilities. The author cosiders the call-type optio represetatio of the guarateed bee t ad determied bouds for the premium payable by the policyholder. I variable auity (VA) ad UL busiess, the cotributio is typically speci ed by the policyholder, although rider risk bee ts are mostly charged for explicitly through risk premiums before ivestmet of the cotributio. The ivestmet guaratee charges are levied from the policyholder s uderlyig fud i the form of asset-based charges, i.e. the charges are expressed as a percetage of the value of the uderlyig fud or sub-accout. I the followig, we derive the coditioal lower boud approximatio for the value of di eret types of embedded optio ad the asset-based ivestmet guaratee charge for these optios. We cosider regular premium VA busiess ad ivestmet guaratee rider bee ts that are cotiget o death ad survival, i.e. the GMDB ad GMAB ivestmet guaratee types. We further show how the coditioal lower boud approximatio ca be used to validate sesitivity measures, the so-called greeks, for hedgig. We assume a Black-Scholes-Merto settig throughout, although it is importat to ote that the coditioal lower boud is a versatile approach ad ca be determied i a model depedet or model idepedet case. A example of the model idepedet case is the recet paper by Che et al. (2008) i which the 4

6 authors ivestigate static super-replicatig strategies for Europea-type call optios writte o a weighted sum of asset prices. For complex models or i some model idepedet cases, the coditioal lower boud might result i a approximatio that is ot aalytically tractable. I such cases ad where a poit estimate su ces, umeric solutios ca be used. Ivestmet guaratee o erigs pose several challeges to isurace rms. The guaratees of idividual policies are largely depedet, although the diversi catio bee t arisig from the various uderlyig fud choices o sets the depedecy to some extet. Isurers will fud the iitial cost of settig up a hedgig portfolio or reservig requiremet at iceptio by recoupig the cost over the policy term through asset-based charges. This icome at risk poses a additioal cost i the price of the guaratee. Fially, the embedded optios of ivestmet guaratees take the form of exotic pathdepedet derivatives ad require multiple ested simulatios whe valuig across stochastic ecoomic scearios. The coditioal lower boud is derived i sectio 2 for a simpli ed acial product, i.e. the bee t cosidered is ot cotiget o death or survival. I sectio 3, we show that the coditioal lower boud theory easily exteds to more sophisticated bee ts ad costig structures by addig typical costs as well as mortality to the simplied acial product cosidered i sectio 2. The popular GMDB ad GMAB / GMMB structures are covered i sectio 3 ad i sectio 4, we propose a possible dyamic hedgig solutio for the aforemetioed structures. 5

7 2 A simple product We rst cosider a simpli ed acial cotract for ease of expositio. Cosider a ivestor who ivests a regular cotributio k, k = 0; 1; :::; ;with a rm. I exchage for these cotributios, the rm guaratees the ivestor the greater of a guarateed miimum bee t of b ad the fud value S at maturity of the cotract, i.e. the bee t payout at time is B = max (b ; S ) : We assume that the ivestor, age x at iceptio, survives the cotract with probability 1, i.e. p x = 1; ad we do ot allow for other decremets such as surreders. Let F k deote the value at time k of oe uit of the fud i which the cotributios are ivested by the rm. The value of the ivestor s portfolio V k+1 at the ed of the period (k; k + 1), before the ext period s cotributio, is give by the recursive formula: V k+1 = [V k + k ] F k+1 F k (1) Equatio (1) states that the value at the ed of a period durig the cotract term is equal to the regular cotributio added to the existig value of the fud ad the accumulated to the ed of the period by the fud growth factor F k+1 F k. We assume V 0 = 0 by costructio. Uder the assumptio that the ivestor survives each period ad that all cotributios have bee paid, we ca express equatio (1) as: V k+1 = kx j=0 j F k+1 F j ; k = 0; :::; 1 (2) 6

8 The miimum guarateed bee t at maturity ca be writte as follows: B = b + max 0; V b (3) or: B = V + max b V ; 0 (4) Equatio (3) describes the guarateed bee t as the value of the guaratee plus a call optio o the fud value at maturity with a strike price equal to the value of the guaratee. Equatio (4), o the other had, describes the guarateed bee t as the fud value at maturity plus a put optio o the fud value at maturity with a strike price equal to the value of the guaratee. The expressios i equatios (3) ad (4) aturally allow for the use of acial ecoomics i the pricig of the guarateed bee t. 2.1 The cocept of comootoicity The cocept of comootoicity, or "commo mootoicity", provides the theoretical framework for the coditioal lower boud approximatio. A recet ad largely self-cotaied overview of comootoicity is give by Deelstra et al. (2010). For the purposes of this paper, the followig de itio of comootoicity su ces. De itio 1 For ay -dimesioal radom vector X = (X 1 ; X 2 ; :::; X ) with multivariate cumulative distributio fuctio (cdf) F X (x) ad margial uivariate cdf s F X1 ; F X2 ; :::; F X ad for ay x = (x 1 ; x 2 ; :::; x ) 2 R, there exists a radom variable Z ad o-decreasig fuctios f i, i = 1; 2; :::;, 7

9 such that: X d = (f 1 (Z) ; f 2 (Z) ; :::; f (Z)) The cocept of comootoicity allows the use of certai properties to arrive at a accurate approximatio of the true distributio of the uderlyig fud ad the cotiget bee ts o the uderlyig fud. The followig de itios will be used i the followig: De itio 2 For a comootoic sum S c of the vector X c = (X1; c X2; c :::; X), c the iverse distributio fuctio of a sum of comootoic radom variables is equal to the sum of the iverse distributio fuctios of the margial distributios: F 1 S c (p) = X i=1 F 1 X i (p) ; p 2 [0; 1] (5) De itio 3 For a comootoic sum S c of the vector X c = (X c 1; X c 2; :::; X c ), the distributio of the comootoic sum S c ca be give i terms of the stoploss premiums of the margial compoets: E X (S c d) + = E (X i i=1 d i ) + (6) where F 1 S (0) d F 1 (1) : The d i; for i = 1; 2; :::;, are give by: c S c d i = F 1( d) X i (F S c (d)) (7) where d 2 [0; 1] is determied by: F 1( d) S (F c Sc (d)) = d (8) 8

10 For the logormal case, the iverse distributio fuctio is left-cotiuous ad strictly mootoic ad the value of d i equatio (8) is therefore equal to oe. The decompositio formula i equatio (6) ca be iterpreted acially such that the value of a Europea call optio o a stochastic sum S c of share prices with strike price d is equal to the sum of the values of Europea call optios o the costituet shares with strike prices d i as determied i equatio (7). I fact, Jamshidia (1989) proved that i the case of oefactor mea-revertig Gaussia iterest rate models, a Europea optio o a portfolio of discout coupo-bearig bod decomposes ito a portfolio of Europea optios o the idividual discout bods i the portfolio. 2.2 Value of the embedded optio For ease of expositio, assume that the parameters of the cotract (b ; k ) are give at iceptio of the cotract, ad that the ivestmet fud price process ff (t) ; t 0g evolve accordig to a geometric Browia motio process with costat drift ad costat volatility, i.e. assume a Black- Scholes-Merto settig: df (t) F (t) = dt + dw (t) ; t 0 (9) with iitial value F (0) > 0; ad where fw (t) ; t > 0g is a stadard Browia motio. Uder a uique equivalet martigale measure Q, see e.g Harriso ad 9

11 Pliska (1981), we have: F (t) = F (0) exp 2 t + W (t) ; t 0 (10) 2 By combiig the results i equatio (2) ad i equatio (10), we d the followig expressio for the fud value at maturity V : V = k exp 2 ( k) + (W () W (k)) 2 (11) The cotiget claims o the fud value i equatio (11) are path-depedet ad take the mathematical form of a arithmetic Asia optio. To show this i the Black-Scholes-Merto settig, we use the time reversal property of Browia motio. This property allows us to state a ew process that maitais the probabilistic structure as the old process, i.e. W ~ () = W () W (k) ;where = k: By substitutig the time reversed Browia motio i equatio (11), we d: V = exp =0 2 + W 2 ~ () = F (12) =0 Assume ow that the cotributios are level across the policy term, so that we have: where k = 0; 1; :::; k = 1 j = 1 (13) j=0 1 ad deotes the total premium paid over the policy term. Combiig the results i equatio (12) ad equatio (13), we 10

12 have: V 1 = F (14) =0 The result i equatio (14) implies that ay cotiget claim o the policyholder s fud will take the form of a cotiget claim o the arithmetic average of the uderlyig fud prices across the policy term, i.e. take the form of a arithmetic Asia optio o the uderlyig fud. The value of Asia optios do ot have aalytical solutios ad various approximatios are available i the form of umerical methods or double itegral type bouds. Rogers ad Shi (1995) suggest a coditioal lower boud i the cotiuous averagig case. Dhaee et al. (2002a) derived comootoic bouds for a arithmetic Asia optio ad illustrated the e cacy of these bouds i a Black-Scholes-Merto settig. I particular, the authors demostrated the icredible accuracy of the coditioal lower boud as a approximatio to the exact value of a arithmetic Asia optio. 2.3 Coditioal lower boud (CLB) approximatio I the followig, we approximate the distributio of a sum of partially depedet radom variables such that true fud value V stochastically domiates the approximatio of the fud V i a covex order sese, but i a optimal way. The coditioal lower boud is de ed to be the expectio of V coditioed o some iformatio variable ; i.e.: V = E V j (15) Ideally, the choice of is such that V ad are as alike as possible. 11

13 This choice reduces to optig for a sigi cat level of depedece betwee V ad. I the followig, we iitially choose such that it is a liear combiatio of the variability of V ; i.e.: = k [W () W (k)] (16) where k is a determiistic costat. The CLB approximatio is optimised by selectig the appropriate value of the cotat k. The CLB approximatio ca also be optimised with respect to the tail of the distributio, e.g. Vadu el et al. (2008) propose locally optimal approximatios i the sese that the relevat tail of the distributio of E [S j] is a accurate approximatio for the correspodig tail of the distributio of S. The variability of the fud value i equatio (12) ad the coditioal radom variable i equatio (16) stem from the Browia motio di ereces, Y k = W () W (k), for k = 0; 1; :::; 1: The distributio of ay Browia di erece Y k give follows a coditioal bivariate ormal distributio. The CLB approximatio that is yet to be optimised therefore follows from equatio (15): V = = k E 1 X F () F (k) j = k e ( )( k) E e Y k j k e ( r 2 k)( k)+r k p kz (17) where r k is Pearso s correlatio coe ciet for the pair (Y k ; ) ad ca 12

14 be foud i the usual way: r k = Cov [Y k; ] Yk where the covariace of the pair (Y k ; ) is give by: Cov [Y k ; ] = l mi( k; l) (18) l=0 The expressio i equatio (17) is ot a optimal boud ad the variace of V eeds to be maximised with respect to i order to optimise the CLB approximatio. The variace of V is give by: V ar V = k l e (2 k l) p p e 2 r k r l k l 1 l=0 (19) By expadig the expoetial term i brackets by a rst order Taylor series, we ca approximate the variace of V by: V ar V 2 r 2 S 2 S (20) where S = P 1 ke ( k) Y k ; ad the correlatio coe ciet of the pair (S; ) is deoted by r S. We ca therefore maximise the expressio for the variace of V i equatio (20) by maximisig the correlatio coe ciet r S : This is oly the case if the pair (S; ) is perfectly correlated, egatively or positively. The optimum choice of is therefore give as: = S = k e ( k) Y k (21) 13

15 Comparig our optimal choice i equatio (21) to our iitial choice i equatio (16) ; it is evidet that the optimal choice remais a liear combiatio of the Browia di ereces Y k but with the costat k de ed as: k = k e ( k) (22) The correlatio coe ciet r k therefore becomes: r k = P 1 l=0 le ( l) mi( k; l) p q P k 1 P 1 j=0 l=0 j l e (2 j l) mi( j; l) (23) 2.4 Value of the embedded optios Now that we have a approximate distributio for the value of the fud V uder the equivalet martigale measure Q, we ca use this distributio to approximate the discouted payo or value of the embedded guaratee of our simple acial cotract. From equatio (17), it is evidet that V is a fuctio of a multiple of the stadard ormal radom variable Z: By usig the expressio i De itio 3, we d: E "! k e r p k kz b + # = E X k F 1 X k F V (b ) + (24) where F X (x) ad F 1 X (p) deote the cumulative distributio fuctio of the radom variable X i the value x ad the iverse of the fuctio i the value p, respectively. X k = k e r p k kz, where k = k e ( rk)( 2 k) : We also de ed the logormal radom variable 14

16 By usig the result of De itio 2, we d a expressio for the iverse distributio fuctio of the CLB approximatio V : F 1 V (p) = F 1 X k (p) = k e r p k k 1 (p) (25) We ow have the tools eeded i order to determie the value of the discouted payo s as give i equatios (3) ad (4). First, cosider the calltype payo i equatio (3) : By usig the results from equatios (24) ad (25), we obtai: = E "! k e r p k kz b " k E + # e r p p k kz r k k 1 e F V! (b ) + # (26) The logormal radom variable X leads to the al result for the value of the call-type payo : h E V b + = k e ( i h p k) r k k 1 F V i (b ) b h1 F V i (b ) (27) The put-type payo i equatio (4) is foud by usig put-call parity, i.e.: h E i h b V = E + V b + i + b E V. (28) 15

17 The result for the value of the put-type payo is give by: h E = i b V + k e ( k) h p r k k + 1 F V i (b ) + b F V (b ) (29) The ukow quatity F V solved by usig the expressio i equatio (25), i.e.: (b ) i equatio (27) ad equatio (29) is k e ( rk)( 2 k) e r p k k 1 F V (b ) b = 0 (30) 2.5 Asset based price of the embedded optio I the previous sectio, the value of the embedded optios at iceptio was determied. I this sectio, we determie the periodic price charged i respect of the embedded optios. A aual maagemet fee is charged o the value of the fud at the ed of each respective year or cotributio period. The charge ca be stated aually but deducted mothly or quarterly. Note that at iceptio the embedded optio has to be purchased by the isurer or a replicatig portfolio has to be set up i order to e ectively maage the risk of the guaratee. The maagemet charges therefore recover the iitial guaratee liability outgo. The aual maagemet fee solutio is a discrete case of the cotiuous pricig solutio, which is a deductio i the yield of the uderlyig fud. Milevsky (2006) gives a structured approach to obtaiig the actual price of the guaratee whe recoupig the cost of the guaratee cotiuously from the uderlyig fud. 16

18 Cosider the asset based charge e deducted aually from the fud V. The fud value at time is give as: V = k (1 e) k e ( )( k)+(w () W (k)) (31) Comparig equatio (31) above to equatio (11) we see that oly the premium vector chages, i.e. the guaratee charge reduces the cotributio that participates i the fud growth. By followig the same steps i derivig the CLB approximatio as i sectio 2.3, we d the value of the optimisig costat k of the iformatio variable as: k = k (1 e) k e ( k) (32) ad the optimal correlatio coe ciet r k as: r k = P 1 l=0 l (1 e) l e ( l) mi( k; l) p q P k 1 P 1 j=0 l=0 j l (1 e) 2 j l e (2 j l) mi( j; l) (33) The value of the embedded call optio at iceptio of the cotract is give as: C 0 = h p k (1 e) k e k r k k 1 e b h1 F V i (b ) F V i (b ) (34) 17

19 ad the value of the embedded put optio at iceptio of the cotract as: P 0 = h k (1 e) k e k +e b F V p r k k + 1 F V i (b ) (b ) (35) Agai, the ukow quatity F V (b ) i the expressios of equatios (34) ad (35) is foud by usig the result i equatio (25), i.e.: k (1 e) k e ( rk)( 2 k) e r p k k 1 F V (b ) b = 0 (36) The aual charge for the guaratee e ca ow be foud by fair value priciples, i.e. equatig the expected preset value of the premium with the expected preset value of the bee ts. For the embedded put optio, we have: k e k = e E V + P0 (37) The ukow guaratee charge e is therefore foud by umerically solvig the followig expressio with respect to e: e b F V (b ) h p k e h1 k (1 e) k r k k 1 F V ii (b ) = 0 (38) The derivatio for the embedded call optio follows i a similar way. 18

20 2.6 Illustrative Example The accuracy of the coditioal lower boud approximatio is illustrated i the followig by determiig the value of the embedded guaratee usig a Mote Carlo method as a proxy of the true value. I Table 1, we calculated the discouted values of the embedded put optios for the simpli ed product. The accuracy of the call optios follow by put-call parity.variace reductio i the form of atithetic variates was used. Simulated results are based o paths for each set of parameter values. We assume a cotiuous risk-free rate of 5% per aum with volatilities of 20%, 30% or 40% per aum. The cotributios k ; k = 0; :::; ; are assumed at 100 per aum ad the term of the cotract is take as = 10 years. We also vary the guarateed amout b by cosiderig the followig ve values (500; 750; 1000; 1250; 1500) : The CLB colum gives the coditioal lower boud approximatio, while the MC ad s.e. colums give the Mote Carlo estimate with associated stadard error. The aual charge was solved for by usig equatio (38) ad is give for di eret parameter values i Table 2. The charges are give i basis poits (bps) of the fud value charged aually. Charges i excess of or approachig 100% are omitted. It is evidet from Table 2 that the charges icrease substatially as the volatility icreases or as the risk-free rate decreases. The sesitivity of the optio value to these parameters ad therefore of the price of the optio eed to be cosidered i a comprehesive risk maagemet strategy. I sectio 4, we cosider ad test a possible hedgig strategy that aims to immuise the 19

21 Table 1: Coditioal Lower Boud (CLB) estimates of the embedded put optio values with = 10, = 5% ad varyig volatilities, compared to Mote Carlo (MC) estimates ad their stadard errors (s.e.). b CLB MC s.e. 20% % % Table 2: Charges i basis poits (bps) for the embedded put optios with = 10 ad varyig volatilities ad risk-free rates. b = 20% = 30% = 40% = 1% = 5% = 10%

22 importat risk factors. 3 Life-cotiget guarateed miimum bee ts I this sectio, we geeralise the simpli ed pure ivestmet product of sectio 2 to allow for mortality risk by cosiderig guarateed death ad guarateed survival bee ts. The premiums for acillary risk bee ts such as dread disease, disability ad premium waiver bee ts are usually deducted from the policyholder s cotributios before the cotributio is ivested i the sub-accout. We assume that the isurer has a risk-eutral positio to mortality risk, i.e. the VA busiess portfolio cosists of a large eough populatio of idepedet policyholders such that the aggregate mortality risk is diversi ed. I the case where mortality risk is fully diversi able, the radom lifetimes of policyholders ca be replaced by their expected lifetimes. The Law of Large Numbers imply that the variability i the expected value of the idepedet radom lifetimes of a cohort of idetical policyholders teds to zero, i.e. the expected lifetime of a cohort of policyholders becomes determiistic. We also assume that mortality risk is idepedet of ivestmet risk, thereby allowig us to express the real world probability of a itegrated risk evet as a the product of the idividual ivestmet risk ad mortality risk evets. Idepedece betwee acial ad biometric risks are ot ecessarily maitaied whe replacig the physical probability measure with the equivalet martigale measure. Dhaee et al. (2010) provides a overview 21

23 of the implicatios of modellig itegrated risks i the combied market of acial ad biometric risks. We assume i the followig that the combied market is complete ad that idepedece betwee the sub-markets holds for the equivalet martigale measure. Cosider a VA product that is uderwritte o a life aged x at time 0 ad who aims to auitise the VA i years. Aual premiums k are used to purchase uits of the opted uderlyig ivestmet fud. A allocatio cost charge c is charged for whe purchasig uits, which implies that (1 c) k worth of uits are available of each cotributio k. We assume for simplicity that ay fractioal uit of the uderlyig fud ca be purchased. The allocatio cost typically covers the itial expeses, commissio or advisory fees ad ogoig allocatio expeses. The allocated premiums are used to purchase uits of the ivestmet fud at the fud s bid price F k at the start of the year (k; k + 1). We assume that the bid price of oe uit of the ivestmet fud at time t is equal to the value of the fud F k, i.e. the ivestmet fud prices are give et of the bid-o er spread. The isurer charges the bid-o er spread by icreasig the o er price at time k, i.e. settig the o er price at F k 1 for some value 0 < < 1: This meas that the policyholder has to buy the uits at a higher price tha the value of the fud. The premium for acillary mortality bee ts is charged as a deductio, the risk premium (r) k, from the regular cotributio. The savigs premium or et cotributio at time k is de ed by: (s) k = (1 ) (1 c) k (r) k (39) 22

24 Assumig that the policyholder is still alive at time k ad that o uits have bee withdraw before time k, the fud value V k+1 at time k+1 is give by: V k+1 = kx j=0 (s) F k+1 j (1 e) k+1 j ; k = 0; :::; 1 (40) F j 3.1 Guarateed miimum maturity bee t We assume that a su cietly large book of idepedet isured lives are held by the isurer. This meas that the probability that a policyholder aged x survivig to time t is replaced by the frequecy of survival of the cohort of the populatio aged x: P [T x > t] = t p x (41) where T x deotes the radom lifetime of a policyholder aged x. The CLB approximatio for the sub-accout after allowig for deductios ad charges is give as: V = (s) k (1 e) k e ( )( k)+(w () W (k)) (42) By followig the same steps i derivig the CLB approximatio as i sectio 2.3, we d the value of the optimisig costat k of the iformatio variable as: k = (1 ) (1 c) k (r) k (1 e) k ( k) e = (s) k (1 e) k e ( k) (43) 23

25 ad the optimal correlatio coe ciet r k as: r k = p k q P 1 j=0 P 1 l=0 (s) P 1 l (1 e) l e ( l) mi( k; l) l=0 (s) j (s) l (1 e) 2 j l e (2 j l) mi( j; l) (44) The preset value at time 0 of the embedded call optio is give by: C 0 = h p x (s) k (1 p e) k e k r k k 1 p x e b h1 F V i (b ) F V i (b ) (45) ad the preset value of the embedded put optio is give by: P 0 = h p x (s) k (1 e) k e k + p x e b F V p r k k + 1 F V i (b ) (b ) (46) I the case of the put-type optio represetatio of the GMMB, the fair value priciple leads to the asset-based charge e to be solved from: kp x (s) k e k = p x e E V + p x P 0 = h p x (s) k (1 p e) k e k r k k 1 + p x e b F V F V i (b ) (b ) (47) Note that the preset value of cotributios, left-had side of equatio (47), excludes the cotributio attributable to acillary risk bee ts ad allocatio charges. For the call-type optio represetatio of the GMMB, 24

26 which equals a -term bod ad a call optio, the applicatio of the fair value priciple results i: kp x (s) k e k = p x e b + p x C 0 = h p x (s) k (1 p e) k e k r k k 1 + p x e b F V F V i (b ) (b ) (48) It is evidet from equatios (47) ad (48) that the solutio for the guaratee charge e is idetical for the call-type optio ad put-type optio represetatios. 3.2 Guarateed miimum death bee t The risk premium (r) k periodic basis, i.e. the risk premium (r) k ca be used to explicitly charge for the GMDB o a is solved for from the expressio: (r) k = A (x + k) e E [S R ] (49) where A (x + k) is the cost-loaded actuarial premium for a oe-year term isurace with death bee t equal to 1 ad sold to the policyholder of age x + k, ad where: S R = max b k+1 V k+1 ; 0 (50) where V k+1 cotais the risk premium (r) k, cf. equatio (39). We assume i the followig that the charge for the GMDB is recouped 25

27 by a asset-based charge e g that forms part of the regular maagemet fee e, i.e. e = e a + e g where e a deote all other asset-based charges such as fud maagemet charges. The call optio ad put optio represetatios remai the same as i equatios (45) ad (46), but we ow eed to solve for the asset-based charge e g by applyig fair value priciples. I the case of the put-type optio represetatio of the GMDB, the asset-based charge e g is solved from the expressio: kp x (s) k e k = kp x q x+k e (k+1) E V k+1 + b k+1 V k+1 + (51) The asset-based charge for other fees besides the ivestmet guaratee charge is assumed to be exogeously give. Therefore, the oly ukow i equatio (51) is the GMDB asset-based charge e g : 3.3 Guarateed miimum death ad survival bee ts Edowmet type VA products typically o er a guarateed miimum bee t o both death ad survival. Sice these life-cotiget evets are mutually exclusive, the ivestmet guaratee bee t simpli es to beig the sum of the two life-cotiget bee ts, i.e.: B 0 = B q 0 + B p 0 = kp x q x+k e (k+1) E + t p x e E max V ; b p max b q k+1 ; V k+1 + (52) where b q k+1 deotes the guarateed miimum death bee t ad bp deotes 26

28 the guarateed miimum survival bee t. By usig the split of the aual maagemet charge, we solve for the ukow asset-based guaratee charge e g from the expressio: kp x (s) k e k = B q 0 + B p 0 (53) Note that we assume oe asset-based guaratee charge for both the death ad survival bee ts. Note that should the isurer guaratee cotributios iclusive of acillary bee t charges ad allocatio charges, the savigs premium (s) k is the replaced by the full cotributio k o the discouted cotributios side. The portio of the total guaratee charge applicable to the death bee t ca be established by: B q 0 B 0 e g (54) ad similarly for the portio of the guaratee charge applicable to the survival bee t: B p 0 B 0 e g (55) 3.4 Illustrative example I the followig, we demostrate the applicatio of the results obtaied for solvig for the asset-based guaratee charge e g : We cosider the guaratee charge for the death bee t ad for the survival bee t for varied ecoomic assumptios. The cotributios k ; k = 0; :::; ; are assumed at 100 per aum with a cotract term of = 10 years. The GMDB, b k+1 ; ad GMMB, 27

29 b ; are assumed to provide retur of cotributios : We also cosider a bee t equal to 50%, 75%, 125% ad 150% of cotributios. The policyholder is assumed to be a male aged 30 at iceptio of the policy with mortality accordig to the PMA92 mortality tables as published by the Cotiuous Mortality Ivestigatio Bureau (CMIB) of the Istitute ad Faculty of Actuaries, U.K. A determiistic mortality assumptio ca result i serious cosequeces to a isurer s solvecy if its book of policies has few policyholders or if the book of policies cosist of largely homogeeous policyholders that are susceptible to the same mortality shocks. I such cases, the Law of Large Numbers does ot apply ad the added mortality risk implies a additioal cost, which must be charged to the policyholder for the isurer to accept the addtioal risk. This cost stems from either more aggressive reisurace treaties or more striget reservig requiremets. A extreme upper boud is cosidered for the embedded optio value i the case of the GMMB. If the uderlyig fud value were to fall to zero, the payo becomes certai ad the oly radomess relates to the survival of the policyholder. The bee t therefore becomes a pure edowmet cotract with a sum assured equal to the certai payo. It is evidet from Table 3 that the value of the embedded optio i the combied market case will approach the extreme upper boud for large values of the volatility parameter ad small values of the risk-free rate parameter, i.e. implyig o time-value of moey. I Table 4, the extreme upper boud to the value of the embedded optio is also give. Similar to the GMMB example, the oly radomess relates to the mortality of the policyholder. Therefore, i this case the bee t would 28

30 Table 3: Coditioal Lower Boud (CLB) results for the discouted values of the GMDB embedded put optios with = 10 for a policyholder aged 30. Pure b 20% 30% 40% Edowmet 1% 50% % % % % % 50% % % % % % 50% % % % % be the term life death bee t with a sum assured equal to the certai payo i the evet of death. Agai, the embedded optio value i the combied market case will approach the extreme upper boud for high volatility values ad low risk-free parameter values. 4 Hedgig strategies 4.1 Itroductio I the followig, we derive sesitivity measures, the so-called greeks, to implemet a dyamic hedgig strategy. A good review of dyamic hedgig strategies ad the various measures used ca be foud i Taleb (1997) ad i Hull (2008). Life compaies might opt for static hedgig strategies, as opposed 29

31 Table 4: Coditioal Lower Boud (CLB) results for the discouted values of the GMMB embedded put optios with = 10 for a policyholder aged 30. Term Life b 20% 30% 40% Bee t 1% 50% % % % % % 50% % % % % % 50% % % % % to dyamic hedgig strategies, should the ivestmet guaratee structure allow it, e.g. recurrig premium busiess with a GMMB requires a portfolio of forward-startig Europea put optios to be hedged accordig to a static hedgig strategy where these might ot be available i the market. I this sectio, we cosider approximate measures for the three greeks typically cosidered by life isurace rms i their dyamic hedgig programmes, i.e. the delta, vega ad rho of a portfolio. It might be possible for a isurer to trasfer its exposure to ivestmet risk to a third party, such as a reisurer or a ivestmet bak, although these markets might prove ifeasible durig ad followig acial turmoil. Reisurace rms typically have a set risk appetite for ivestmet busiess ad do ot readily take o the ivestmet risk, while Ivestmet baks ca 30

32 o er a structured product to the isurer to trasfer some or all of the ivestmet risk or aid i the maagemet of a existig hedgig programme. The applicatio of usig existig market istrumets, icludig over the couter istrumets, to set up a static hedgig portfolio is formally cosidered by, for example, Che et al. (2008). 4.2 A Proposed Hedgig Solutio I order to derive the delta at time t of the embedded optios, we eed to express the values of the embedded optios i terms of the fud price at time t; where t is de ed o the cotiuous iterval (0; ) : I the previous sectios, we assumed that time elapses i a discrete way, i.e. that the time variable k measures the discrete momets i time k = 0; 1; :::; 1: I the followig, we rst derive a expressio for the value of the embedded optio at some istataeous time t; where 0 t ad the d the delta measure by takig the derivative of the derived expressio with respect to the uderlyig fud price at time t, F t : I the follwig, we cosider oly the put-type embedded optio sice the derivatio for the call-type optio follows i a similar way. I our dervivatio of the CLB approximatio, the iitial fud value F (0) cacelled out. Assume ow the ltratio F t ad k < t, we therefore have: F () F (k) = F (t) F (k) exp 2 ( t) + (W () W (t)) 2 (56) The CLB approximatio for the fud value V is give as: 31

33 V X 1 = where g k (Z) is de ed as follows: (s) k g k (Z) (57) 8 >< g k (Z) = >: F (t) 1 F (k) e( 2 2 rt 2 p )( t)+r t tz for k < t e ( r 2 k)( k)+r k p kz for k t (58) By followig the same steps i derivig the CLB approximatio as i sectio 2.3, we d the value of the embedded put optio at time t as: P t = dte 1 X (s) h k F (t) F (k) k=dte (s) k e (k t) h p r t t + 1 p r k k + 1 F V F V i (b ) i (b ) + e ( t) b F V (b ) (59) where the value of F V (b ) ca be foud i the usual way, viz.: F 1 k g k (Z) F V (b ) b = 0 (60) The delta measure of the embedded put optio of the GMMB i the combied market is give by: t (t) = tp x dte 1 X (s) h k F (k) p r t t + 1 F V i (b ) (61) The delta measure of the GMDB embedded put optio, assumig that 32

34 the policyholder is alive at time t, is give as: t = j=dte 1 jp x q x+j dte 1 X (s) h k F (k) p r t t + 1 F V j+1 i (b j+1 ) (62) The delta measure of a embedded optio i the combied market is therefore a weighted sum of the costituet delta measures, where the weights are determied by the probability of the evet that drives the bee t, i.e. survival or death. The vega ad rho of the embedded optio aim to immuise the portfolio with respect to the model parameters ad r, respectively, thereby mitigatig model risk. Although the delta measure was foud aalytically, a umeric ite di erecig approach could have bee used. A poit estimate approximatio of the embedded optio value allows up ad dow shifts i the embedded optio value to be calculated with speed ad ease. The vega ad rho of the embedded optio measure chages i the embedded optio value due to chages i the model parameters ad are ot aalytically tractable. Therefore, the vega ad rho are calculated usig the ite di erecig approach. A commo approach is to cosider both a up ad a dow shift i the optio value, see e.g. Wilmott (2006). The vega for the put-type optio is calculated as: V P t ( + ) P t ( ) 2 (63) 33

35 where P t () deotes the put-type optio value as a fuctio of the give parameter value ad deotes the magitude of the small shift i the parameter value. Likewise, we d the rho for the put-type optio as: t P t ( + ) P t ( ) 2 (64) The magitude of the shift i the parameter value depeds o the rm s risk maagemet framework ad is usually a fuctio of expected future volatility. 4.3 Illustrative Example Cosider a ivestor who cotributes a mothly cotributio of 1 at times, k = 0; 1; :::; ; for a term of 5 years, i.e. = 60. For simplicity, we assume rst that all paymets are paid with certaity ad that death is the oly decremet, i.e. o lapses or surreders. To assist the illustratio, a i-themoey 200% retur of cotributios guaratee o the ivestmet is assumed, i.e. a strike price of b = 120. A sigle market sceario was simulated from a Black-Scholes-Merto ecoomy with parameters = 20% ad = 5%. A delta hedgig strategy is cosidered by usig both the aalytical delta measure of sectio 4.2 ad the delta measure foud by ite-di erecig, i.e.: t P t [F (t) + F (t)] P t [F (t) F (t)] 2F (t) (65) The GMMB is illustrated i Figure 1, while the GMDB is illustrated 34

36 Embedded Optio Value / Hedge Value Figure 1: Compariso of performace of delta measures for GMMB Embedded Optio Value Hedge Value (Explicit Formula) Hedge Value (Fiite Differecig) Duratio (Moths) i Figure 2. I both gures, the embedded optio value is illustrated by the solid lie, while the dotted lie represets the approach of equatios (61) ad (62) ad the dashed lie represets the approach of equatio (65) : 5 Coclusio The complexity of isurace cotracts set i the combied market of acial risk ad mortality risk pose sig cat challeges to the risk maagemet frameworks of isurers. The coditioal lower boud approximatio aids i addressig the three questios posed by Hardy (2003). It allows quick feasiblity studies of product structures durig product developmet, ad estimates of the cost of capital i the case of reservig or the cost of a suitable hedgig programme i the case of dyamic or static hedgig. The use of the coditioal lower boud approximatio further allows isurers to verify the 35

37 Embedded Optio Value / Hedge Value Figure 2: Compariso of performace of delta measures for GMDB Embedded Optio Value 5 Hedge Value (Explicit Formula) Hedge Value (Fiite Differecig) Duratio (Moths) reasoableess of estimates arisig from more complex ad resource itesive pricig ad reservig models. The coditioal lower boud approximatio ca be exteded to allow for more complex asset pricig models ad mortality models. Refereces Aase, K., Persso, S., Pricig of uit-liked life isurace policies. Scadiavia Actuarial Joural, Baciello, A., Ortu, F., 1993a. Pricig equity-liked life isurace with edogeous miimum guaratees. Isurace: Mathematics ad Ecoomics 12, Baciello, A., Ortu, F., 1993b. Pricig guarateed securities-liked life i- 36

38 surace uder iterest-rate risk. I: Proceedigs of the 3rd ICA AFIR Colloquium, Rome. Vol. I. Black, F., Scholes, M., The pricig of optios ad corporate liabilities. Joural of Political Ecoomy 81, Boyle, P., Hardy, M., Reservig for maturity guaratees: two approaches. Isurace: Mathematics ad Ecoomics 21, Boyle, P. P., Schwartz, E. S., Equilibrium prices of guaratees uder equity-liked cotracts. The Joural of Risk ad Isurace, 44, Brea, M., Schwartz, E., The valuatio of america put optios. Joural of Fiace 32, Brea, M., Schwartz, E., Pricig ad ivestmet strategies for guarateed equity-liked life isurace. Hueber Foudatio Moograph Series. Wharto School, Uiversity of Pesylvaia. Che, X., Deelstra, G., Dhaee, J., Vamaele, M., Static superreplicatig strategies for a class of exotic optios. Isurace: Mathematics ad Ecoomics 42, Deelstra, G., Dhaee, J., Vamaele, M., A overview of comootoicity ad its applicatios i ace ad isurace. I: Øksedal, B., Nuo, G. (Eds.), Advaced Mathematical Methods for Fiace. Spriger, Heidelberg, Germay. Delbae, F., Equity liked policies. Bulleti de l Associatio Royale des Actuaires Belges 44,

39 Dhaee, J., Deuit, M., Goovaerts, M., Kaas, R., Vycke, D., 2002a. The cocept of comootoicity i actuarial sciece ad ace: Applicatios. Isurace: Mathematics ad Ecoomics 31, Dhaee, J., Deuit, M., Goovaerts, M., Kaas, R., Vycke, D., 2002b. The cocept of comootoicity i actuarial sciece ad ace: Theory. Isurace: Mathematics ad Ecoomics 31, Dhaee, J., Kukush, A., Schoutes, W., A ote o the idepedece betwee acial ad biometrical risks. Workig Paper. Hardy, M., Ivestmet Guaratees. Joh Wiley & Sos Ic., Hoboke, New Jersey. Harriso, J., Kreps, D., Martigales ad arbitrage i multiperiod securities markets. Joural of Ecoomic Theory 20, Harriso, J., Pliska, S., Martigales ad stochastic itegrals i the theory of cotiuous tradig. Stochastic Processes ad their Applicatios 11, Hull, J., Optios, futures, ad other derivatives, 7th Editio. Pearso Pretice Hall. Hürlima, W., April - Jue Aalytical pricig of the uit-liked edowmet with guaratee ad periodic premium. Pravartak 3, Jamshidia, F., A exact bod optio formula. Joural of Fiace 1,

40 Merto, R., Theory of ratioal optio pricig. Bell Joural of Ecoomics ad Maagemet Sciece 4, Milevsky, M., The calculus of retiremet icome: acial models for pesio auities ad life isurace. Cambridge Uiversity Press. Møller, T., Risk-miimizig hedgig strategies for uit-liked life isurace cotracts. ASTIN Bulleti 28, Møller, T., Hedgig equity-liked life isurace cotracts. North America Actuarial Joural 5, Nielse, J., Sadma, K., Equity-liked life isurace: a model with stochastic iterest rates. Isurace: Mathematics ad Ecoomics 16, Nielse, J., Sadma, K., Uiqueess of the fair premium for equityliked life cotracts. The Geeva Papers o Risk ad Isurace: Theory 21, Nielse, J., Sadma, K., Pricig bouds of asia optios. Joural of Fiacial ad Quatitative Aalysis 38, Rogers, L. C. G., Shi, Z., The value of a asia optio. Joural of Applied Probability 32, Shrager, D., Pelsser, A., Pricig rate of retur guaratees i regular premium uit liked isurace. Isurace: Mathematics ad Ecoomics 35,

41 Simo, S., Goovaerts, M., Dhaee, J., A easy computable upper boud for the price of a arithmetic asia optio. Isurace: Mathematics ad Ecoomics 26, Taleb, N., Dyamic hedgig: maagig vailla ad exotic optios. Joh Wiley & Sos Ic., Hoboke, New Jersey. Thompso, G. W. P., Fast arrow bouds o the value of asia optios. Tech. rep., Cetre for Fiacial Research, Judge Istitute of Maagemet, Uiversity of Cambridge. Vadu el, S., Che, X., Dhaee, J., Goovaerts, M., Herard, L., Kaas, R., Optimal approximatios for risk measures of sums of logormals based o coditioal expectatios. Joural of Computatioal ad Applied Mathematics 221, Wilmott, P., Paul Wilmott o Quatitative Fiace, 2d Editio. Vol. 3. Joh Wiley & Sos Ltd., Chichester, West Sussex, Eglad. 40

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