Batten freesample

8/14/2019 Batten freesample

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PREFACE

The d isc ip lin e o f life c on tin ge nc ie s h as fo r man y g en era tio ns d efin ed th e b asic p rin cip le s o f th ew ork o f th e actu ary to a far g reater extent th an any oth er sing le su bject, tech nic al or o the rw ise.M any actu arial stu den ts, th eir emplo yers, and their professo rs ha ve con sidered the moment ofsuccess on the Society of Actuaries exam ination covering life contingencies as a point ofmetamo rp ho sis from th e sta tu s o f n eo ph yte to th at o f a fu ll-fle dg ed p ro sp ec t fo r membe rsh ip a ndmean ingfu l con tr ib uti on s i n h is o r h er cho sen p ro fe ss ion.Every profession deserves a single reference which seeks to define its m ost fundam entalp rin cip le s, a rti cu la te ly and w it h te chni ca l p re cis io n. The act ua ria l p ro fe ss ion is fo rt un at e to h avesuch a reference in Actuarial Mathematics (to be referred to herein as "the text") as anu nq ue stio ne d a uth ority fo r a ll w ho se ek membersh ip . It is a n e njo ya ble b ut a lso c ha lle ng in g te xtfrom w hich to teach, as m any enlightening digressions m ust be m ade in the classroom w hichexpand s tuden ts ' though t p rocesses beyond the mathematica lly-prec is e theo ry and i llus tr at ions.H av in g work ed in a fa ce -to -fa ce c la ssro om env iro nmen t w ith more th an 2 50 0 stu de nts who h av epas sed the relevant Society o f Actuar ie s o r Casual ty Actua rial Society Examina tion , one conc lusionh as b ec ome c le ar. R eg ard le ss o f its q ua lity , th e stu de nt w ho stu die s o nly th e p re sc rib ed te xt is a t as er ious competi tive d is advan tage . I ts exe rc is es do no t bear a c lose res emblance to those cons truc tedb y th e Examin atio n Committe es. Its d eriv atio ns, th ou gh math ematic ally so un d, a re o fte n o f su chleng th and te chni ca li ty t ha t th eir tr ue s ignif ic an ce is masked . P erh ap s it cou ld even be a rgued th at i tsh eavy emph asis on statistics cloud s some of th e relationsh ips w hich are truly more log ical th anstatistical.T his vo lume, no tw ithstanding its su pplem entary con cepts, its form ulas, its techn iqu es, and itsexe rc ise s and det aile d s olu tio ns , s hould never b e con sid ered a s a s ub stit ute f or th e text its el f. Thi sis stric tly a su pp lem ent w hich is no t self-con tained and w ould be fou nd to b e of little v alue in th ea bse nc e o f a n a cc ompany in g d eta ile d stu dy o f Actuaria l Mathematics .Much of the material contained herein was developed while instructing students of lifecon ti ng encie s a t Geo rg ia S ta te Un iv er si ty . Upon comp le tio n o f t he cou rse s equence, con si stent lyov er 90% of the stud ents w ere su ccessfu l o n the exam o n their first attem pt. T he ex ercise s, w ithfew exc ep tio ns, a re o rig in al p ro blems to whic h th ese y ou ng p eo ple w ere e xp ose d th ro ug ho ut th eircourse of study. They w ere not created for exam -passing purposes, but rather to provide thestudents a source of questions designed to test, and to expose their w eaknesses on, the basicpr incip les of l ife cont ingencies .The form at of this m anual should assist the student from his or her first introduction to the m aterialth rou gh th e o ften fran tic d ay s o f fin al p rep aratio n. E ach ch ap ter b eg in s w ith su pp lem en tary co nceptsand examples designed to shed light on those topics which the reader of Bowers may not haveassim ilated sufficiently. A t the end of each chapter are practice problem s follow ed by solutions.These 360 problems should be used, a few at a time, to test knowledge on topics as they aretr ad itio na lly e xamin ed b y th e a ctu aria l s oc ie tie s.Fo llowing the chap te r summaries i s a s ec tion consi st ing o f solut ions to 87 careful ly chosen ques tionsfrom the B ow ers te xt. B efore a ttempting the tex t e xercises, it is recommende d that th e stu den tcarefu lly read the introd uction to this sec tio n on P age 23 9 in this m anual. F inally, four tw en ty-

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VI

question practice exam s are included, preferably to be taken under sim ulated exam conditions in thefinal few weeks leading up to the exam date. Upon scoring the results and analyzing the givenso lu tio ns fo r an y careles s erro rs o r altern ate techn iq ues, th e stu dent sh ou ld b e p rep ared fo r w hatev erw ill be encountered on exam day.Spe cia l th an ks g o to G ail H all, F SA , MAAA , V ic e P re sid en t o f P ro du ct D ev elo pmen t a t ACTEX,who originated the idea of developing a 2005 version of this m anual and whose guidanceth ro ug ho ut th e p ro ce ss w as in va lu ab le . T he ty pe se ttin g w as p erfo rmed b y Marily n B ale sh isk i o fACTEX w ith g re at p atie nc e, g oo d h umor, a nd d ilig en t a tte ntio n to d eta ile d n ota tio n a nd symbolsuni qu e to lif e con tin genc ie s. My g ra ti tu de goe s to P ro fe ss or Hal P eder sen, ASA, o f t he Unive rs ityo f Manito ba , fo r h is c are fu l re ad in g o f th e man usc rip t w hic h c re ate d th e o pp ortu nity fo r e xp an de dex pla nations in ad dition to a final se arch for erro rs. I am also g ratefu l to the stu dents w ith w hom Ihave w orked over a 40 year period in their pursuit of m astery of the subject. M any of the ideasand te chnique s con ta in ed here in we re re fin ed by v ir tu e o f th eir in cis iv e que stio ns and comments .M y greatest level of appreciation goes to m y wife, Jane, who perform ed m uch of the originaltyping on this manual. M ore importantly, she has been both a tireless supporter and a truein sp ir ati on to me th roughout th is and ear li er p ro je ct s.P ro ofrea din g o f m aterial such as this can n ever b e complete in sp ite of m any effo rts to ferret o ute ve ry m isp la ce d su pe rsc rip t o r c oe ffic ie nt. T he e rro rs whic h rema in a re th e fu ll re sp on sib ility o fth e a uth or; it is h op ed , a t le ast, th at n on e o f th ese e rro rs a re o f a su bsta ntiv e n atu re .Life contingencies exam inations are passed far m ore readily by candidates w ho approach the topiclo gic ally a nd who emp lo y in sig htfu l te ch niq ue s th an b y th e a cc omp lis he d math ematic ia n/s ta tis tic ia nwho treats it as a challenge to derive overcomplicating formulas under rigorous examinationco nd itio ns. T his v olume rep resen ts an effort to co nv in ce ex am in atio n cand id ates th at th is o bserv atio nis accurate. Historically, many who have depended primarily on mathematical skills in theirp reparatio n h av e d on e so to th eir own d etrim en t.

With th e lo gic al a pp ro ac he s in co rp ora te d in th is man ua l, su pp lemente d b y in te nsiv e p ra ctic e inp ro blem-so lv in g, th e stu de nt o f life c on tin ge nc ie s sh ou ld n ot o nly b e su cc essfu l o n th e first e xamattem pt but should also m axim ize his or her enjoym ent of a subject w hich on the surface m ayseem depressing. Over the past forty years, even as m y personal om ega loom s closer, I haveincreasingly enjoyed working with students of life contingencies. It is m y wish that thise njo ymen t w ill b e e vid en t to th e re ad er.

B ig Canoe , Geo rg iaJ anuary 2005 R obert W . B atten, FSAP ro fess or Emeritu s o f A ctu arial S cien ceGeorgia S ta te Un iv er si ty


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Chapter S ixSupplementary Concepts1. Whereas only tw o distinct "principles" are applied to prem ium determ ination in C hapter S ix,

th ere a re m any s itu atio ns in whic h ne ith er m ay be a pplied . T he s tu den t must b e ke en ly awa re th atb y fa r th e more impo rta nt o f th ese , th e eq uiv alen ce p rin cip le , m ay n ot b e a ss um ed u nle ss at lea sto ne o f th e f ollowin g is in dic ate d:1) th e s ta tem en t th at E[L] = 0 , or sim ply that th e equ ivalen ce prin ciple (E P) is to b e im posed.2) th e u se o fth e word s "b en efit p remium,"3) the presence of a "punctuated" prem ium sym bol, such as Px, P(Ax), nPx, or P(nlax)'

U se of the generic "P " o r ju st the word "p remium" g iv es no in dica tio n o f th e prin cip le ,if a ny , whic h is b ein g a ss umed .Note th at s ettin g up a p rem ium e qu atio n o f the fo rm "ac tu aria l p re se nt v alu e of fu tu re prem iums "equals "actuarial present value of future benefits" is inappropriate in the absence of anassump tion o f the equ ival ence p rinc ip le .

2. T he other nam ed principle, although introduced in C hapter S ix, is rarely used in practice ando ft en p ro du ce s ma jo r in co ns is te nc ie s a nd absurd itie s. The " pe rc en tile p rem ium p rin cip le " c an b es hown to p ro du ce p remium s in some c irc um sta nce s whic h are th e s am e fo r te rm in su ra nc es as fo re ndowmen t in su ra nc es , o r e ven p rem iums equ al to z ero fo r c erta in te rm in su ra nc es .It s ho uld b e c le ar th at th e f ollowin g phra se s a re e qu iv ale nt a nd , h en ce , in te rc hang eable .1) "Find the sm allest annual prem ium which an insurer m ay charge in order to incur ap ositive lo ss on n o more th an 1 5% o f its p olic ies ."2) "Find the f if te en th per cent il e p remium."

Exercises 6-4, 6-5, and 6-27 illustrate questions of this type as well as the recommendedapproaches.


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86 Chapter 63. T he variances of prospective loss random variables L are closely related to those of present value

random variables for single premium insurance policies introduced in Chapter Four. Thevariance formulas are very compact where the equivalence principle is assumed, but are not assim ple or as easy to rem em ber in the absence of the equivalence principle.Example 1

If L = vT - Pa T I ' T ~ 0, then V AR [L] = 2Ax - (Ax)2 if th e(1 - Ax)2equ iv alence p ri nc ip le is a ss umed, i.e ., if P = P (A x). Otherwise,

V AR [L] = [2Ax - (Ax)2] (1 + f) 2.

Note that each of these variance formulas shows clearly that V AR [L] > V A R [Z],where Z is the present value random variable for continuous w hole life insurance inChapte r Fou r. Thi s r ef le ct s t he g re ate r " unce rta in ty " i nh erent i n L becau se t he p remiumsa re c on tin uo us fo r life ra th er th an b ein g p aid in fu ll a t t = O.

Example 2If L = vT - PaT!' 0::; T ::;n= vn - Pa;-r, T > n,then 2A --(A-iV AR [L] = x:nl x:nl( 1 - A _ )2x:nlif th e equ iv alence p ri nc ip le i s a ss umed, i .e ., if P = P (Ax: ;- r) .Otherwise,

V AR [L] = [2Ax:;-r - (Ax:;-r)2] (1 + ~)2.

Ana lo go us fo rmula s e xist fo r fu lly d isc re te who le life a ndendowmen t insurance los s r andom variab les.

4. U nfortunately, sim ple variance form ulas sim ilar to those just stated do not exist for loss randomvariables associated with products other than whole life or endowment insurance. W henvariances of such random variables are required, they are best obtained through an appeal tob asic p rin cip le s o r th ro ug h a te ch niq ue to b e in tro du ce d in Chapte r E ig ht.


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Chapter 6 87

5. The pure endowment annual prem ium P x:~ e qu als th e re cip ro ca l o f th e a ctu aria l a cc umula te dvalue 8 x:;;-j . This is b ecause the sh are of the su rv iv or w ho h as d eposited P x: ~ a t t he b eg inningof each year for n years is the contractual $1 pure endowment, i.e.,

P 1-8 - - 1x :n l x :n l - .

6. Level benefit prem iums such as nP x ' P~:;;-j'an d P x:; ;- jpurchase identical insurance betw een agesx and x + n. The difference in the magnitude of the prem iums is solely attributable to thein vestm en t featu re o f th e co ntract.A ccordingly, com parisons of the policy values of survivors at age x + n m ay be effected byana ly zing fu tu re b enef it s; p ast b en efi ts may be igno red becau se t hey a re equ iv al en t.

Examples P_pl-(P pi) .. A n x x:nln x - x : ; ; - j S x:;;-j = x+ n = P 1-x:nlP -- P(P - - P) 8 - - 1 - A - x:nl n xx:nl n x x:nl - x+ n - p 1-x:nl

P - - pi -( p - - pi - ) 8 - - 1 - x:nl x:nlx:nl x:nl x:nl - - P 1-x:nlB ecause of the form of the right-hand sides of the above relationships, the approach to thesetyp es of p ro blem s is often refe rred to as th e "P m inu s P ov er P " tech niq ue.These concepts will be extended after the introduction of the notion of reserves in ChapterSeven.

7. The tw o m ost useful identities in life contingencies relate the m ost basic quantities in ChaptersFour, Five, and Six to each other. W hile virtually every student w ill be com fortable w ith therelationshipA-=l-


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Ax:1i] = 1 - d a x : 1 i ] Px:1i]Px:1i] + da = 1 - Ax:1i] 1x:1i] d Px:1i]+ d

dA -~-d- = x:nlx::nl 1 - Ax:1i] a -x:nl

- 1 - 8ax:1i] P(Ax:1i])A --x:nl - P(Ax:1i]) + 8a - - 1 - Ax:1i] 1x:nl - 8 P(Ax:1i]) + 8

8A - ~-8(Ax:1i]) - x:nl- I-A - a -x:nl x:nl

88 Chapter 6

8. If L = vT - P(Ax)aTf' T ~ 0, and if the forces of mortality and interest are f.L and 8,re sp ec tiv ely , th e p ro ba bility th at th e in su re r w ill re aliz e a g ain (i.e ., a n eg ativ e lo ss) o n a ra nd omin su re d is g iv en b y

(~ )M/6f.L+8 .Com pare this statem ent to Supplem entary C oncepts #17 in C hapter Four and #10 in C hapter Five.

Special Mortalitv L aw sI. C onstant Force of M ortality

Px = vq x = P~:1i]-- --IP(Ax) = f.L = P(Ax:1i])

For f ul ly d is cr ete whole li fe ,V AR [L] = p . 2Ax = pqq + 2i + i2

i f t he equ iv al en ce p rin ci pl e is a ss umed.For f ul ly con tin uous whol e l if e,

V AR [L] = 2Ax = f.L-: 28i f t he equ iv al en ce p ri nc ip le is a ss umed.


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Chapter 6 89

II. U niform D istribution of D eathsP(Ax) = * Px

P (AI ) ip lx:;:;j = {; x:;:;jP eA ) ipl- + P 1-:;:;j = {; x:nl x:nl

p(m) = Pxx a(m) - (3(m)(Px + d)P-p(ml = x:nlx:nl a(m) - (3(m)(P;:;:;j+ d)

p(m) - nP xn x - a(m) - (3(m)(P~:;:;j + d)

p(m) (AI_)= i. pi 0:)h x:nl {; h x:nl

Exercises6-1

6-3

Rank in in cre asi ng nume ric al s iz e:i)ii )iii)iv )

P(Ax)P (Ax)P(4)(Ax)P(2)(Ax)

6-2 Letp(x) = 90~x,0::;x


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Chapter 6 95

Solutions6-1 Key Approach: Since each of the prem ium s involves a num erator of Ax, we must analy ze th e

denominator s wh ich are , r espect ively, ax, ax, a~4)an d a~2).Wh ile fo rmula s fo r th ese a nn uitie scould be compared, it is m ore instru ctive for the stu dent to co nsider their relativ e m agn itud eslogically.First, ax makes th e fu ll an nual payment at th e beginnin g o f each year; hence it is the largest, i.e.,mo st expen siv e, o f th e annuity v al ue s.Second, a~2)exceeds a~4),as it m akes larger paym ents sooner in each respective year.Finally, a(4) exceeds ax , w hich can be thou ght of as a(oo), for the same reason that a(2) exceedsx x x..(4 )a .xThus, s in ce th e p remiums a re inve rs ely re la ted to t he ir cor re sponding annui tie s, we have

(i ) < (iv) < (iii) < (ii) .

6-2 K ey A pproach: Since the m ortality basis here is DML , and since our basic tools for handlingin su rances invo lv ing DML wer e developed i n Chapt er Fou r, we s hould u se th e r ela tio ns hipdA -p - = x:nJx:nl 1 - A -'x:nl

A60:iO[ - A 1 - + lOE6060:101- aiO[ + 1020- 30 v 30- .6176.

h fi P .6176dT ere ore 60:101= 1 - .6176 = .0914


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96 Chapter 6

6-3 Key Technique: When tw o different prem ium rates are being analyzed for a loss randomvariable L, it is u su ally e ffic ie nt to c on sid er a ra tio when c omp arin g th e tw o v aria nc es.Key Formulas: V AR [L] = (1 + % ) 2 e Ax - (Ax)2)

-- AP(Ax) = _xax(FCWL)

Letting L * be the loss random variable when the premium is 1f, we have

VAR[L*] - (1+~)2 - ( 1f+8 )2 - 16.V AR [L] - (1 + P~Ax)r - P(Ax) + 8(1f+8)2(ax)2 = 16(1f + 8) ax = 4

Since 8 = 1 - Ax = .05, we h av eax1f = 16 - .05 = .20.

6-4 a) Key Concept: Since lo = 1 00 an d ls = 75, and since theinsurer m ust be a "w inner" in at least 75% of the cases, it m ustearn a profit fo r all de aths o ccu rring after age 5 , i.e., o n a ll po liciesfo r w hich d eath o ccu rs in the six th yea r o r la te r.." . 1 f8 6f.0 4 > 1 0,0 00.".1f > 1449.634

b) Key Concept: Since lo = 100 and ls = 75, and since theinsurer cannot tolerate a loss in as m any as 25% of the cases, itcan no t be a "lo ser" o n tho se p olic ies fo r w hich d eaths o ccu r in thefifth po licy y ear. T hu s it m ust a t least b re ak e ve n on th e p olic iesw hich b ecome d eath c laim s in th e fifth y ear..". 1f851.04 ~ 10,000." . 1f ~ 1775.261

N otes: T he student should understand clearly the fine, but significant, difference in the tw oparts of this question. Further, it should be clear why each answer must be rounded ]JQ,p roduci ng fi na l v al ue s o f $1449 .64 and $1775 .27, r es pecti ve ly .


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In troducto ry Note

A s is o ften the case w ith m athema tica l te xtb oo ks w hich lean towa rd the theo retical, ActuarialMathematics (Bowers, et al) contains m any end-of-chapter exercises w hich are of lim iteda pp lic ab ility in p re pa ra tio n fo r a p ro blem-so lv in g p ro fe ssio na l e xamin atio n su ch a s S oc ie ty o fA ctu arie s Exam M .T his d oes no t m ean that p ro ofs and d erivation s can no t be o f immen se value to actuarial stu den tsin their attem pt to m aster the theory and application of life contingencies. Every textbookexercise w hich a student com pletes, w hether it be sophisticated or elem entary, sym bolic ornumerical, h ard or e asy, surely add s to ove rall u nd erstand ing eve n if p erhap s in som e cases o nlymarginally.N ev erth ele ss, some exe rc ise s a re o bv io usly o f mo re b en efit th an o th ers. In th is se ctio n th e re ad erw ill find solutions to 87 carefully selected questions from the B ow ers text w hich seem to havespecial significance to the actuarial student of life contingencies. W hile it is hoped that thestu den t w ill attem pt oth er text ex erc ises as w ell, a s a m inim um these 8 7 qu estions an d solution ssh ou ld b e c le arly u nd ersto od b y a ny se rio us c an did ate fo r Exam M .S ev era l a dd itio na l o bse rv atio ns sh ou ld b e mad e. F irst, th ese e xe rc ise s sh ou ld n ot b e a ttemp te d(n or th e so lu tio ns re ad ) u ntil o th er se ctio ns o f th is man ua l h av e b ee n c are fu lly stu die d. T his stu dyshould include m astery of special m ortality law s (such as deM oivre's L aw , constant force ofmo rta lity , a nd Modifie d D eMo iv re 's L aw ). S ec on d, w hile tra ditio na l so lu tio ns to e xe rc ise s su chas these rely heavily upon calculus (especially integral calculus), the student should m ake aser ious a tt empt to minimize, and perhaps near ly e liminate, calcu lus-based solut ions . H istor ical ly ,p ro fe ssio na l e xamin atio ns in life c on tin ge nc ie s re qu ire in te gra l c alc ulu s in n o more th an o ne o rtwo questions per exam , although m any others could be solved using calculus but often inu nn ecessarily lo ng er and less in tu itive w ays. Y ou w ill no tice th at ve ry few solution s co ntain edh ere in in vo lv e in te gra tio n a nd many empha siz e lo gic al, le ss so ph istic ate d te ch niq ue s. T hird , th estudent should attem pt each of these exercises in a serious m anner before looking at the givensolution s. If a first effo rt p ro duc es a co rrect a nsw er, b ut is o btain ed in a relativ ely in efficientm an ner, o nly the n w ill the stude nt be able to gain m aximum b enefit from th e solu tions co ntain edhere.T oo m any students of the B ow ers text have ignored its exercises and, in m any cases, even thetext itself. O ften this approach has been to the student's great detrim ent. W hile it m ay havew orked for som e, it has been an unw ise decision for m any others. T he concepts illustrated bym ost of these 87 problems have appeared in sim ilar form on the SOA Course 150 Exam andSOA/CAS Exam 3 over the last decade. The student who has not studied them or whoapp ro ac he s th em w ith p ure ly math ematic al te ch niq ue s h as m isse d o ut o n a g old m in e o f v alu ab lemate ri al for exam preparat ion .


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6.6

6.7

6.8

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Solutions to Sele cte d T ex tb oo k E xe rcis es

J1 ( J1 )22 Ax - (A x )2 - Ji+'M - --;;+8(1 - Ax)2 - (J1~8 t2 Ax - (Ax)2Somewhat tedious a lg eb ra p roduce s th e re su lt = 2Ax = J1(I-Ax)2 J1+28

Key Fac ts :- - 1P(Ax) = =--6ax- 0 Of ' 0x = ex 1 1 =

.. W ith 6 = i = 0,P(Ax) = J-ex

Key Obse rv atio ns: F or c on tin uo us who le life (C ha pte r 4 ),2- - 2VAR[Z] = Ax -(A x) .

For f ull y con tin uous whole li fe (Chap te r 6 ),2 Ax - (Ax)2VAR[L] = (I-Ax)2 .

Since Ax is b etw een 0 an d 1, clearly VAR[L] > VAR[Z], prov ing the asser tion .

20110Ax20 PC2o llO Ax ) =ax:201

Al -A Ix:3Ol x:201ax:201

Al -A I20 P1:3O]- P1:201 = x:~ x:2Ol

ax:201

C learly, the identity is show n to be correct.

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Batten freesample

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