TO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC

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1 TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies segmet (MLC). This versio is iteded for the eam offered i May 007 ad thereafter. Please be aware that the DVD semiar does ot deal with the Fiacial Ecoomics segmet (MFE) of Eam M. The purpose of this memo is to provide you with a orietatio to this semiar, which is devoted to a review of life cotigecies plus the related topics of multi-state models ad the Pois process. A 6-page summary of the topics covered i the semiar is attached to this cover memo. Although the semiar is orgaized idepedetly of ay particular tetbook, the summary of topics shows where each topic is covered i the tetbook Models for Quatifyig Risk (Secod Editio), by Cuigham, Herzog, ad Lodo. I order to be ready to write the MLC eam, you eed to accomplish three stages of preparatio: This first stage is to obtai a uderstadig of all the uderlyig mathematical theory, icludig a mastery of stadardized iteratioal actuarial otatio. This DVD review semiar is desiged to eable you to obtai that uderstadig. The secod stage is to prepare, ad the master, a complete list of all formulas ad relatioships (of which there are may) eeded for the eam. For those who do ot wish to prepare their ow lists, a very complete set of formulas (364 i total) is available from ACTEX i the form of study flash cards. The third stage is to do a large umber of eam-type practice questios, begiig with the edof-chapter eercises is the tetbook. I additio, you should purchase oe (or more) of the several eam-prep study guides that have bee prepared for that purpose. A relatively small umber of sample problems (34) are worked out for the group i this DVD. A copy of these questios, with detailed lutios, is attached to this cover memo for your referece. Good luck to you o your eam

2 A. Parametric Survival Models Summary of Topics. Age-at-Failure Radom Variable X (3.) a. CDF b. SDF c. PDF d. HRF e. CHF f. Momets g. Actuarial otatio ad termiology. Eamples (3.) a. Uiform b. Epoetial c. Gompertz d. Makeham e. Weibull f. Others via trasformatio 3. Time-to-Failure Radom Variable T (3.3) a. SDF b. CDF c. PDF d. HRF e. Momets f. Discrete couterpart K g. Curtate duratio K() 4. Cetral Rate (3.4) 5. Select Models (3.5) B. The Life Table. Defiitio (4.). Traditioal Form (4.) a. b. d ad d c. p ad p d. q ad q 3. Other Fuctios (4.3) a. ad t b. PDF ad momets of X --

3 c. m q d. PDF ad momets of T e. Temporary epectatio f. Curtate epectatio g. Temporary curtate epectatio h. Cetral rate 4. No-Itegral Ages (4.5) a. Liear assumptio b. Epoetial assumptio c. Hyperbolic assumptio 5. Select Tables (4.6) C. Cotiget Paymet Models. Discrete Models (5.) a. Z, ad its momets b. Z : ad its momets c. Z, ad its momets; covariace with d. Z: ad its momets; covariace with e. Z :, ad its momets. Group Determiistic Iterpretatio (5.) 3. Cotiuous Models (5.3) a. Z, ad its momets b. : c. Z : Z ad Z, ad their momets ( m ), d. Z ad its momets e. Evaluatio uder epoetial distributio f. Evaluatio uder uiform distributio 4. Varyig Paymets (5.4) a. B i geeral b. (IA) c. ( IA) : d. ( IA), e. ( I A), ad ( IA) : ( DA) : : ad ( DA) : : ( I A), ad ( DA), 5. Approimatio from Life Table (5.5) a. Cotiuous models b. m thly models Z : Z : --

4 D. Cotiget Auity Models. Whole Life (6.) a. Immediate b. Due c. Cotiuous. Temporary (6.) a. Immediate b. Due c. Cotiuous 3. Deferred (6.3) a. Immediate b. Due c. Cotiuous 4. m thly Paymets (6.4) a. Immediate b. Due c. Radom variables d. Approimatio from life table 5. No-Level Paymets (6.5) a. Immediate b. Due c. Cotiuous E. Fudig Plas. Aual Paymet Fudig (7.) a. Discrete paymet models b. Cotiuous paymet models c. No-level fudig. Radom Variable Aalysis (7.) d. Preset value of loss radom variable e. Epected value f. Variace 3. Cotiuous Paymet Fudig (7.3) a. Discrete paymet models b. Cotiuous paymet models 4. m thly Paymet Fudig (7.4) a. Discrete paymet models b. Cotiuous paymet models 5. Icorporatio of Epeses (7.5) -3-

5 F. Reserves. Aual Paymet Fudig (8.) a. Prospective method b. Retrospective method c. Additioal epressios d. Radom variable aalysis e. Cotiuous paymet models f. Cotiget auity model. Recursive Relatioships (8.) a. Group determiistic aalysis b. Radom variable aalysis cash basis c. Radom variable aalysis accrued basis 3. Cotiuous Paymet Fudig (8.3) a. Discrete paymet models b. Cotiuous paymet models c. Radom variable aalysis 4. m thly Paymet Fudig (8.4) 5. Icorporatio of Epeses (8.5) 6. Fractioal Duratio Reserves (8.6) 7. No-Level Beefits ad Premiums (8.7) a. Discrete models b. Cotiuous models G. Multi-Life Models. Joit-Life Model (9.) a. Radom variablet y b. SDF c. CDF d. PDF e. HRF f. Coditioal probabilities g. Momets. Last-Survivor Model (9.) a. Radom variable T y b. CDF c. SDF d. PDF e. HRF f. Momets g. Relatioships of Ty ad T y -4-

6 3. Cotiget Probability Fuctios (9.3) 4. Cotiget Multi-Life Cotracts (9.4) a. Cotiget paymet models b. Cotiget auity models c. Premiums ad reserves d. Reversioary auities e. Cotiget isurace fuctios 5. Radom Variable Aalysis (9.5) a. Margial distributios b. Covariace c. Joit fuctios d. Joit-life e. Last-survivor 6. Commo Shock (9.6) H. Multiple-Decremet Models. Discrete Models (0.) a. Multiple-decremet tables b. Radom variable aalysis. Competig Risks (0.) 3. Cotiuous Models (0.3) 4. Uiform Distributio (0.4) a. Multiple-decremet cotet b. Sigle-decremet cotet 5. Actuarial Preset Value (0.5) 6. Asset Shares (0.6) I. Pois Process. Properties ( ). Miture Process (.3.4) 3. Nostatioary Process (.3.5) 4. Compoud Pois Process (4.) -5-

7 J. Multi-State Models. Review of Markov Chais (Appedi A). Homogeeous Multi-State Process (0.7.) 3. Nohomogeeous Multi-State Process (0.7.) 4. Daiel s Notatio (SN: M-4-05) -6-

8 Practice Questios. Let a survival distributio be defied by is 60, fid the media of X. S ( ) a b, for 0 k. If the epected value of X X. Give that k, for all 0, ad 0 p 35.8, fid the value of 0 p If X has a uiform distributio over (0, ), show that m,.50m for. 4. Give that e 0 5 ad, for 0, fid the value of Var( T0 ), where T0 deotes the future lifetime radom variable for a etity kow to eist at age Give the UDD assumptio ad the values , , ad , fid the value of q Let k deote a costat force of mortality for the age iterval ( k, k). Fid the value of e :3, the epected umber of years to be lived over the et three years by a life age, give the followig data: k k e k e k Give the followig ecerpt from a select ad ultimate table with a two-year select period, ad assumig UDD betwee itegral ages, fid the value of.90 q [60].60. [ ] [ ] 60 80,65 79,954 78, ,37 78,40 77, ,575 76,770 75, Calculate the value of Var( Z5), give the followig values: A A.004 i A5 A50 p

9 9. Let the age-at-failure radom variable X have a uiform distributio with 0. Let f ( z) deote the PDF of the radom variable Z40. Calculate the value of f (.80), give al that.05. Z Z 0. (a) Show that ( IA) A E ( IA). (b) Calculate the value of ( IA) 36, give the followig values: ( IA) 3.7 A.9434 A 35 35:.300 p Assumig failures are uiformly distributed over each iterval (, ), A give the values i., q.0, : ad q.0. calculate the value of. Fid the value of A, give the values a 0, a 7.375, ad Var( a ) 50. T 3. Let S deote the umber of auity paymets actually made uder a uit 5-year deferred whole life auity-due. Fid the value of Pr( S 5 a ), give the followig values: a 4.54 i.04.0, for all t :5 t 4. Show that, uder the UDD assumptio A i ( id) a. 5. Calculate the probability that the preset value of paymets actually made uder a uit 3-year temporary icreasig auity-due will eceed the APV of the auity cotract, give the followig values: p.80 p.75 p.50 v Calculate the value of 000 P( A: ), assumig UDD over each iterval (, ) followig values: ad the A:.804 E.600 i For a uit whole life isurace, let L premium is chose such that E[ L ] 0, ad let variable whe the premium is chose such that the value of * ( ). Var L deote the preset value of loss radom variable whe the * L * deote the preset value of loss radom E[ L ].0. Give that Var( L ).30, fid -8-

10 8. If the force of iterest is ad the force of failure is ( ) for all, show that Var[ L( A )], with cotiuous premium rate determied by the equivalece priciple. 9. Cosider a 0-pay uit discrete whole life isurace, with epese factors of a flat amout.0 each year, plus a additioal.05 i the first year oly, plus 3% of each premium paid. Fid the gross aual premium for this cotract, give the values a 0, a :0 0, ad d Calculate the value of 000( V V ), :3 :3 give the followig values: P.335 i :3. A -year term isurace of amout 400 is issued to (), with beefit premium determied by the equivalece priciple. Fid the probability that the loss at issue is less tha 90, give the values P.8585, V.0445, ad i.0. : :. A 0-pay whole life cotract of amout 000 is issued to (). The et aual premium is 3.88 ad the beefit reserve at the ed of year 9 is Give that i.06. ad q 9.06, fid the value of P Let deote the accrued cost radom variable i the th year for a discrete whole life isurace of amout 000 issued to (40). Calculate the value of Var( K40 0), give the followig values: i.06, a a a Let 0 L( A ) deote the preset value of loss at issue for a fully cotiuous whole life cotract issued to (). Fid the value of 0 V ( A ). give the followig values: 0 0 Var[ L( A )].0 A.30 A A -year edowmet cotract issued to () has a failure beefit of 000 plus the reserve at the ed of the year of failure ad a pure edowmet beefit of 000. Give that i.0, q.0, ad q., calculate the et level beefit premium. 6. Let T ad Ty be idepedet future lifetime radom variables. Give q.080, qy.004, t p t q, ad t py t qy, both for 0 t, evaluate the PDF of Ty at t

11 7. Let T80 ad T 85 be idepedet radom variables with uiform distributios with 00. the probability that the secod failure occurs withi five years from ow. Fid 8. A discrete uit beefit cotiget cotract is issued to the last-survivor status ( ), where the two future lifetime radom variables T are idepedet. The cotract is fuded by discrete et aual premiums, which are reduced by 5% after the first failure. Fid the value of the iitial et aual premium, uder the equivalece priciple, give the followig values: A.40 A.55 a The APV for a last-survivor whole life isurace o ( y), with uit beefit paid at the istat of failure of the status, was calculated assumig idepedet future lifetimes for ( ) ad ( y) with costat hazard rate.06 for each. It is ow discovered that although the total hazard rate of.06 is correct, the two lifetimes are ot idepedet sice each icludes a commo shock hazard factor with costat force.0. The force of iterest used i the calculatio is.05. Calculate the icrease i the APV that results from recogitio of the commo shock elemet. 30. The career of a 50-year-old Profesr of Actuarial Sciece is subject to two decremets. Decremet is mortality, which is govered by a uiform survival distributio with 00, ad Decremet is leavig academic employmet, which is govered by the HRF () y.05, for all y 50. Fid the probability that this profesr remais i academic employmet for at least five years but less tha te years. () () 3. Fid the value of p (), give q.48, q.3, q.6, ad each decremet is uiformly distributed over (, ) i the multiple-decremet cotet. (3) 3. Decremet is uiformly distributed over the year of age i its asciated sigle-decremet table () with q.00. Decremet always occurs at age.70 i its asciated sigle-decremet () table with q.5. Fid the value of (). q 33. Evets occur accordig to a Pois process with rate per day. (a) What is the epected waitig time util the teth evet occurs? (b) What is the probability that the th evet will occur more tha two days after the 0 th evet? 34. Durig a certai type of epidemic, ifectios occur i a populatio at a Pois rate of 0 per day, but are ot immediately idetified. The time elapsed from oset of the ifectio to its idetificatio is a epoetial radom variable with mea of 7 days. Fid the epected umber of idetified ifectios over a 0-day period. -0-

12 Solutios to Practice Questios. We have S(0) b ad S( k) ak 0. Thus a, k 3 S( ). The k k E[ X ] ( ) k S y dy k k k k 60, 0 3k 3 3 k 90, ad thus a. Fially, S( m) m, which leads to m 45. m 4050, ad. Recall that Here ad p 0 35 t t t 0 r dr 0 y dy. p e e.8 e e ky dy k(45) k (35) 400k p 0 40 e ky dy e, e e k (60) k(40) 000k 400k.5.5 e (.8) If X is uiform, the ad t p S( t) t, S( ) that t p t. From Equatio (3.6) we have m 0 tpt dt 0 tp dt 0 dt dt 0 t. ( ).50 0 dt t dt 0 The m.50 m m, as required. --

13 4. The survival model is uiform. The fact that e o 0 5 tells us that 50. The T 0 (0, 40), its variace is (40) is uiform over q80 5. First we use q80 q8.06. Arbitrarily let , to lve for q80.0. Similarly we fid q8.04 ad 8 (.98)(000) 980, 8 (.96)(980) , ad 83 (.94)(940.80) The q ( ) 80.5 ( ) We start with o 3 e :3 t 0 p dt 0 t 0 t 0 t p dt p p dt p p p dt. Uder the costat force assumptio, But t 0 0 t t ( p ) p p dt ( p ) dt. l p l p p p l p e ad 0 e, o e e :3 e e e e e.9754 (.95)(.9744) (.95)(.9493)(.9730)

14 7..90 q[60] q[60].60 (.40 q[60].60 ).50 q[60].40 q[60].40q[60].50q.60q [60].60q [60] [60] Here ad q q [60] [60] [60] 79, , 65 [60] 6 78, , 79,954 [60] (.40)( ) q (.60)( ).90 [60].60 (.40)( ) (.50)( ) (.60)( ( )(.50)( ) Recall that A50 vq50 vp50 A5. The A A A v q v p A which implies A Similarly, A ( vp ) v q A ,.0.0 A5 A50 A5 v q50 v p50 A5 A 5 ( v p 50 ) v q 50 which implies A A , (.0) (.0) The Var( Z ) A A.475 (.6099)

15 T 40, Sice Z v we have we have the trasformatio F l Z ( z) S z. 40 T 40 But ST ( t) 40 t p40 t t sice.05. ad fially l z F ( ) l Z z z, The f Z ( z) d F ( ), 40 Z z 40 dz 3.50z f (.80) Z (3.50)(.80) t z v t l z. The trasformatio is decreasig uder a uiform distributio with 0. Therefore, 0. (a) From Equatio (5.5) we have k k k k k k k k k ( IA) k v q v q ( k ) v q. Let r k k r. The r r r ( IA) A r v q r r r A v p r v q A E ( IA). (b) Note that A35: v The E35 v p35 (.9434)(.9964) Usig the result from part (a), we have The ( IA) A E ( IA) ( IA) 35 A ( IA) E

16 . The relatioship give by Equatio (5.64b) holds for the secod momet fuctios usig a iterest rate based o. We have i A.0 : A : 4 (.) (.90)(.0) l(.) (.) (.).40( ) Recall that A A Var( at ) Var( Y ). We kow that ad The A a 0 A a (4.75 ) (0 ) Var( Y ) , which lves for.035. Fially A (.035)(0) The force of mortality is costat, t p t e for all t. The we ca calculate a v p v p e e e e ( ) ( ) e ( e ) ( ) [.0l(.04)] e e , The that a a a :5 S 5 a if 7 paymets are made, which occurs if ( ) survives to age. This meas ()(.0) Pr( S a ) p e

17 4. Recall that A i A, ad A d a, we have A i ( d a ) i id a i i d a. 5. The APV of the cotract is APV v p 3v p ()(.90)(.80) (3)(.90) (.80)(.75) The preset value of paymets actually made is v.80 if oly two paymets are made, ad.80 3v 5.3 if all three paymets are made. The for the preset value of paymets actually made to eceed the APV, survival to time t is required, the probability of which is (.80)(.75) We have We calculate a : 000 A: (000)(.804) 000 P A:. a a from where, i tur, we fid A: : : A A E a : from : : A d : i A.05 : E A: , l(.05), which gives us ( ) l(.05) :.00. The.05 A ad fially A : A : E , : a 5.0 (000)(.804) 000P A:

18 7. The coditio E[ L ] 0 implies P P. The P d A A d a Var( L ) A A.30, A A d a (.30)( ). The observe that * E P P L A d d ( da ) P P d d d a P P P a.0, d d ( P d) a.0, or P d.0. a Now observe that P d P d (.30)( d a ) d.0 (.30)( d a ) d a * Var( L ) A A (.0) (.30) From Eample 7.5 we kow that P P( A ), ad from earlier results we kow that ad A A. -7-

19 The from Equatio (7.4a) we have Var L( A ) ( ) ( ) ( ) ( ) ( ) 3 3 ( ), as required. 9. The epese-augmeted equatio of value is G a A.05.0 a.03 G a. :0 :0 The G A.05.0a (.03) a :0 (.04)(0).05 (.0)(0) (.97)(0) The value of 000 V :3 is best calculated retrospectively as 000 V :3 000 P :3 v q E The value of 000 V :3 E (.06) (.90).90 is best calculated prospectively as 000 V 000 A P The the differece is :3 : :3 000 (.06)

20 . The loss at issue, deoted L, will be L 400 (400)(.8585) if failure occurs i the first year, which happes with probability q. L The loss will be if failure occurs i the secod year, which happes with probability p q. Certaily the loss will be less tha 90 if ( ) survives to age, we ca coclude that the loss is less tha 90 with probability p. To fid p we first use V v q : , which lves for q.5. The we use : P q p q p.5 p.0 (.0).0 (.0) p p.0.0 which lves for p.83. (.0)( p ).5p (.0).0 p.8585,. Usig the recursive relatioship we have ( VP)( i) 000 q V p, ( )(.06).6 V (.98738), 0 which lves for 0V At duratio 0 the cotract is paid up, the reserve prospectively is 000A The A 0 P a /.06-9-

21 3. From Equatio (8.39) we have Here we have Var( K 0) v ( V ) q p. V a5 000 V a To fid p50 we use a 50 v p50 a 5, The p 50 ( a 50 )( i) (.669)(.06) a Var( K 0) (.06) ( ) (.99408)(.0059) Let L deote 0 L( A ). or Recall that A Var ( L ) ( A A ) (.30 A ).0, a (.30 A ).0( A A ).0 A.40 A.0 0, which lves for A.50. The 0 a0 A 0 V ( A ) a A Whe the failure beefit is a fied amout plus the beefit reserve, we use the recursive relatioship approach. For the first year, where 0 V 0, we have P( i) q (000 V ) p V 000 q V, V P( i) 000 q P(.0) 00. For the secod year we have ( V P)( i) q (000 V ) p V, V ( VP)( i) 000q P(.0) 00 P (.0) 0 P s 0..0 But, prospectively, V 000, as we have P a

22 6. The SDF of Ty is for 0 t. p p p (.080 t )(.004 t ) t y t t y The the PDF is give by t.0003 t, 3 d t p y (.084) t.008 t, dt ad the PDF at t.50 is ()(.084)(.50) (.008)(.50) We seek the value of q Alteratively, p 5 80: :85 p p p : q q q : The APV of the beefit is A A A A. The APV of the premium stream is P a.75 P( a a ) P a.75 P(a a ) We use the give values of P(.50 a.50 a ). A ad a to fid d, from A.40 d a 0 d, d.06, ad the to fid a from The a A.45 d P A A.50 a.50a ()(.40).55 (.50)(0.00) (.50)(7.50).0. --

23 9. Uder the assumptio of idepedece, the APV is A y A Ay Ay Recogitio of the commo shock hazard meas that y.06 as before, but the joit hazard rate is ow * * y y Now the APV is Ay , the differece is 30. We seek the value of ( ) ( ) 5 p 50 0 p 50. We have 5 ( ) () () (.05)(5) 5 p50 5 p50 5 p50 e.7009, 50 ad 0 e ( ) (.05)(0) 0 p50 ( ) ( ) 5 p 0 p.485, Recall that ( ) () () (3) q q q q.96, ( ) p.04. The from Equatio (0.6), () ( ) q () ( ) q /.48/.96 p p ( ) (.04) I the sub-iterval (,.70), Decremet caot occur, Decremet is operatig i a sigle-decremet eviromet ad is uiformly distributed. Therefore ().70 q (.70)(.00).070. If we assume a arbitrary radi of ( ) 000, the we have 70 decremets i the iterval we have 930 survivors at age.70. The there are (930)(.5) 6.5 occurreces of Decremet at age.70, ad therefore i (, ], the probability is () q

24 33. (a) The waitig time for the teth evet, deoted S0, 0 ad. The has a gamma distributio with parameters E[ S 0 0] 5. (b) The iterarrival time for the eleveth evet, deoted T, parameter. The has a epoetial distributio with 4 Pr( T ) e e Note that the process coutig the umber of ifectios occurrig is a stadard (homogeeous) Pois process with 0 (per day), the epected umber of ifectios would be 00 i a 0-day period. But we wish to cout the umber of idetificatios i the 0-day period, ot the umber of actual ifectios. Let t 0 deote the start of the 0-day period. 0 t 0 t 0 (0 ) / 7 For a ifectio occurrig at time t, the probability of beig idetified by time 0 is e t, sice the time-to-idetificatio has a epoetial distributio with mea 7. Thus the rate fuctio for the process coutig all occurrig ifectios is the costat ( t) 0, but the rate fuctio for the process coutig oly ifectios that get idetified by time 0 is the o-costat (0 t) / 7 ( t) 0( e ), which idetifies this process as a ostatioary (ohomogeeous) Pois process. The the epected umber of idetificatios i the iterval from t 0 to t 0 is give by 0 m(0) ( t) dt (0 t) / 7 0( e ) dt / 7 t / 7 0 dt 0 e e dt / 7 t / t 0 e (7 e ) 00 0(.3965)(7)(4.773)

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