Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve noaion. The auhor has, herefore provided replacemen wording for his secion. He has also correced 2 errors. Replacemen wording is noed in red in he aached.
POLICY RESERVES 167 V PV{Fuure Claims} PV{Fuure Ne Premiums} i i i p ic i p i P i i, (6.1) which can also be seen o be where i PV Fuure annual differences beween claims and ne premiums i i i i p [ C P ] (6.2) i p probabiliy of survival o i of policy issued a age, i i C claim cos for someone age a duraion i,issued in year, and P ne premium for someone age a duraion i,issued in year. (1 annual ineres rae) presen value facor, Noe in he above formulae he addiion of, he calendar year of issue. This is a deparure from he annoaion of mos acuarial es, bu is a necessary disincion for policies like major medical, subjec o calendar-based claim rends. Such rends cause he year i claim cos, for a given issue age, o vary from year o year. This is generally no needed for coverages where claims do no vary significanly because of calendar year. For ease of undersanding, his and oher formulae have assumed boh premiums and claims occur a he beginning of he year. A rigorous developmen would accoun for his, and is relaively easy o include in he formulas. Formula 6.1 calculaes a reserve a ime per original issued policy. This is he reserve calculaion one migh use for asse share purposes, or o projec fuure aggregae reserves. If one waned o calculae a reserve s per remaining or per surviving policy, which I will call V hen he formula would be s V i p v i i C i p v i i P i i (6.1a)
168 CHAPTER 6 Formula 6.1a migh be used o calculae reserve facors, which are applied o surviving policies each fuure year, o calculae sauory reserves. These mehods are mos useful in non-inflaionary coverages such as DI and LTC. These formulae are furher complicaed if p, C, or P are on a duraional basis, as will happen, for eample, if eiher moraliy or lapse raes vary by duraion. When 0, Formula 6.1 is 0 i i (6.3) V p C p P i i i i i0 i0 Bu, by definiion, 0 V 0 (reserves sar a ero), so i i i p ic i p i P (6.4) i0 i0 which happens o be he defining condiion of any ne premium 0 P : ha is presen value equals he presen value of claims a ime ero. Formula 6.1 is he prospecive formula for ne level policy reserves, where he ne premiums are assumed o be a consan percenage of he gross premiums. (This is he usual assumpion in such calculaions.) From basic acuarial mahemaics, we know here are equivalen prospecive formulae (like Formula 6.1) and rerospecive formulae. The rerospecive formula corresponding o 6.1a is: 1 1 s 1 1 V i i P i ic i0 i pi v i0 i pi v (6.5a) Rerospecive formulas can be helpful for valuaion acuaries and audiors, in calculaing and checking reserve facors. 3 They are no ofen used for oher purposes. 3 Jordan, Cheser Wallace Jr.. Life Coningencies. pp 115-116.
POLICY RESERVES 169 The aggregae reserves held a ime are he produc of (1) reserve facors shown in 6.5a, muliplied by (2) he number of surviving policies (which, in he cone of calculaions for a single policy, is p ). Since: Then he aggregae reserve is i pi i p p (6.5b) 1 i V i p v i P ic (6.5) i0 EXERCISE 6.1: Using he above formulae, he claim cos sream included in File 12 - Chaper 6 Daa for Eercises.ls workshee, ab 6.1 Daa of he CD-ROM included wih his e, and he discoun raes included in he same spreadshee, consruc he reserve sream shown in ha spreadshee. For he purpose of his eercise, rea claims as occurring a he end of he year and premiums a he beginning of he year. COMPLICATIONS AND VARIATIONS TO THE RESERVE FORMULA The basic formula can be complicaed in a number of ways. The major complicaion, however, is how epenses are refleced in reserve calculaions. Since early epenses (mosly a issue) are ypically higher han laer epenses, he double whammy of having o pay hose epenses plus sar holding policy reserves can pu a large srain on he company s boom line ha year. Sauory and GAAP accouning handle his in differen ways. In sauory accouning, here is no an eplici recogniion of epenses in policy reserves. However, by allowing he use of modified reserve mehods, here is an implici recogniion. The NAIC s model law 4 seing reserve bases allows for wo year full preliminary erm (2YFPT) reserves for coverages oher han LTC, and 1YFPT for LTC. Concepually, he 2YFPT basis can be hough of as allowing he insurer, as hey sar holding reserves, o rea he policy as if i was acually issued wo years laer han i was, wih he policyholder acually wo years older. Algebraically, his ranslaes o having no policy reserves for he firs wo years of he policy. In formulaic noaion: 4 op. ci., Healh Insurance Reserves Model Regulaion.
170 CHAPTER 6 2PT i2 V p i2 p v ic2 i2 i2 for 2, and = 0 for 0,1 i2 i 2 p v i P 2 (6.6) If we are defining V as a reserve per surviving policy, similarly o Formula 6.1a, hen 6.6 becomes: 2 PT, s i2 V i 2 p v ic2 i2 i2 i2 p v i P 2 i2 (6.6a) This ariculaion of he reserve also lends iself o duraional calculaions; 2 PT, s i 2 V[ ] i 2 p[ ] v ic[ ] 2 i2 i2 i2 i 2 p[ ] v i P[ ] 2 (6.6b) The 1YFPT basis has a similar concepual basis. EXERCISE 6.2: Consruc he wo year preliminary erm reserve sream corresponding o he ne level reserves of Eercise 6.1. Assume a 5% increase in he ne level premium for a 37-year old male compared o a 35 year-old male. Many imes in pracice, especially for medical coverages, calculaions are done direcly on elecronic spreadshees, wihou reference o commuaion funcions or published ables. DI and LTC reserves, on he oher hand, are ofen based on published or colleced daa sources, and may rely on formulae in heir developmen.
POLICY RESERVES 171 In GAAP accouning, epenses are eplicily refleced. The policy reserve is calculaed using ne premiums as previously described. If he gross premium srucure has fuure rae changes buil ino i, ne premiums are assumed o change proporionaely o gross premiums, and a i facor reflecing his change would be included in he sream of P. (This doesn mean ha fuure claim rends are assumed beyond he curren raing period hey are usually no. The growh in gross premiums is generally ha which occurs wihou recogniion of fuure increases in he rae schedule iself. I migh acually be a beer heoreical reamen of his o make reasonable assumpions of fuure claim growh and premium growh, however his need is now being me hrough gross premium reserve calculaions, discussed laer.) This reserve is hen referred o as he benefi reserve, and noaion for he variables relaed o i are ypically endowed wih a pos-superscriped B, such as V. To recognie epenses, a parallel calculaion is done using epenses raher han benefis. This epense reserve is acually an asse, bu is concepually equivalen o a negaive reserve, in ha i performs he opposie funcion of benefi reserves. Benefi reserves cause a company o se aside funds which would oherwise become profi, so ha hose funds will (appropriaely) be used laer in he policy lifeime o subsidie laer coss. Epense reserves allow a company o pospone recogniion of cerain epenses, and hus allow funds o flow hrough o profis earlier han hey would if all epenses were fully refleced a he ime hey are incurred. The equivalen of he reserve in epense erms is called he deferred acquisiion cos, or DAC asse. The DAC is composed of deferrable epenses, which are hose incurred o acquire he business. Typically i includes he cos of selling, underwriing, and issuing he policy. (There is someimes also an epense reserve for mainenance epenses, which migh increase over ime, so are similar in naure o he benefi reserves. When recognied, hey are ofen included as a loading on he benefi reserves.) If we denoe he deferrable epenses a ime, from a policy in year o a policyholder aged as E, hen he DAC reserve a ime can be described as in Formula 6.7. Noe ha AV is a erm used o represen he accumulaed value of pas values, in his case wih ineres and erminaions. I is he rerospecive equivalen of he prospecive case s PV, on a per surviving policy basis (in order o calculae facors). B
172 CHAPTER 6 s DAC AV{Deferrable Epense} AV{Ne Epense Premium s} 1 1 1 i 1 i E v i E v i P i 0 i pi i 0 i p i (6.7) This is he epense analogue of Formula 6.1a s benefi reserve. Formulas 6.1 and 6.2 also have analogous epense reserve formulae. In working wih reserves, i is helpful o undersand how one year flows ino he ne. To sar his, i is imporan o keep in mind he difference beween he year-end reserve for a year and he beginning reserve for year 1. The end of year reserve is called a erminal reserve. I is fairly sandard, in doing financial saemen valuaions, o aggregae policies of a given issue age and duraion, and assume a uniform disribuion of issues hroughou he year. This resuls in he average reserve (ypically in he form of a reserve facor o be muliplied by he appropriae eposure value) being he arihmeic mean of he wo erminal reserves. This is called a mid-erminal reserve mehodology. In ha magical momen beween he end of one year and he beginning of he ne, our mehods assume he ne premium has been paid, and he reserve jumps by he value of ha ne premium. Then, during year 1, he reserve accumulaes wih ineres (raising he value of he reserve), and claims all occur a he end of he year (lowering he value of he reserve). The reserve ha resuls from hese calculaions will mach he reserve recalculaed hrough he presen value formulae, because hey are algebraically equivalen. EXERCISE 6.3: Show how formula 6.1 a ime, when adjused by he accumulaion described in he prior paragraph, is equivalen o formula 6.1 a ime 1. EXERCISE 6.4: Using he epense and oher daa from he corresponding workshee in he CD included wih his e, calculae he sream of DAC shown in he resuls ab of he spreadshee. PURCHASE GAAP When a company is acquired, he GAAP effec wih respec o an eising block of policies is ha he DAC asse, which represens he amoried value of he selling company s original coss, is released. (This makes sense, because he DAC is used o align he iming of acquisiion epenses wih fuure premium or profi. When he business is sold, fuure profi is
POLICY RESERVES 173 capialied ino he sales price, so he DAC is as well.) In is place, he acquiring company creaes a VOBA (value of business acquired). The VOBA is he pre-a value of he acquired business. (To he een here are deferrable epenses incurred afer he acquisiion, such as second year commissions higher han hose in years 3+, here will be a DAC asse generaed by his, independenly of he selling company s DAC.) VOBAs are usually calculaed as he presen value of cash profis over he fuure lifeime of he business assumed, discouned a a risk rae of reurn. (This is ofen based on he models used in he appraisal work done o deermine purchase price for he business.) The VOBA is hen amoried over he fuure lifeime of he business, using principles similar o hose described earlier. PRE-FUNDING TRENDS Anoher complicaion o he basic policy reserve calculaion is caused by he various sources of increasing claim coss. In medical insurance here are hree sources for increasing claim coss over ime: inflaionary and similar secular rends, aging of he insured, and duraional rends. The inflaionary and relaed secular rends can be hough of as environmenal rends hose which would cause increases in coss from year o year wih an idenical and unchanging insured risk from year o year. In oday s marke and wih ongoing sieable claim rends for he foreseeable fuure, i is no feasible o compleely pre-fund he hree sources of claim cos increases. Therefore, hose insurers who pre-fund have funded only some par of ha increase. Mos ypically, hey pre-fund aging of he insured and a few years of duraional deerioraion, bu no claim rend. Fuure rae increases are hus assumed o be sufficien o cover he increased cos due o claim rend, furher duraional deerioraion, and he muliplier/cross produc leveraging of hose elemens of cos increases. In recen work, an American Academy of Acuaries ask force indicaed ha i would be unrealisic o assume carriers pricing mehods implicily use more han he firs five years of duraional deerioraion. (Some members believed he acual number o be less han five, for mos carriers.) Mos major medical premiums do no pre-fund rends, bu insead use premiums calculaed o eacly mach ha year s claims. Mos DI and LTC producs are priced on a level premium basis, recogniing aging and duraional effecs.
174 CHAPTER 6 A decade or so ago, in a paper proposing a new regulaory reserve sandard 5, a new modified reserve mehodology was suggesed, which creaed a modified ne premium sream. Tha ne premium sream could be considered as replacing a realisically projeced fuure sream of claim coss wih a ne premium sream ha increases wih secular rend plus %. Wih 0%, he mehod fully prefunds aging, duraion, and he muliplicaive compounding elemen of he combinaion of he hree sources of increasing coss. Premiums are assumed o grow each year proporionaely o secular rend. As increases in value, he level of prefunding goes down, and more of he growh in coss is borne by fuure premium increases. The reserves for such a formula would be calculaed similarly o Formula 6.1, bu perhaps where boh he fuure claims and premium grow annually by some consan value j: i i (1 ) (1 ) i i V i p v ic j i p v i P j i i (6.8) Formula 6.8 is, once again, a calculaion of reserves per issued policy, raher han a reserve per surviving policy, which would be s i i V i p v ic (1 j) i i i i p v i P (1 j) i (6.8a) A furher assumpion in ha paper was ha assumed lapses be recognied in i p. When acual lapses are in ecess of ha amoun, he policy reserves per remaining policy could be increased o recognie he implici aniselecion in ha added lapsaion. This is only one of many such schemes which migh be consruced for such pre-funding, each based on a differing paern for he premium sream. Anoher scheme was proposed by Bob Cumming and Leigh Wachenheim in heir aricle, A Simplified Mehod for Calculaing Conrac Reserves. 6 5 Bluhm, William F. Duraion Based Policy Reserves, Transacions of he Sociey of Acuaries, 1993, pp 11-31 6 Sociey of Acuaries, Healh Secion News, Issue No. 35, June, 1998.