ACTEX. SOA Exam MLC Study Manual. With StudyPlus + Fall 2017 Edition Volume I Johnny Li, P.h.D., FSA Andrew Ng, Ph.D., FSA

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1 ACTEX SOA Eam MLC Sudy Manual Wih SudyPlus SudyPlus gives you digial access* o: Flashcards & Formula Shee Acuarial Eam & Career Sraegy Guides Technical Skill elearning Tools Samples of Supplemenal Tebook And more! *See inside for keycode access and login insrucions Fall 7 Ediion Volume I Johnny Li, P.h.D., FSA Andrew Ng, Ph.D., FSA ACTEX Learning Learn Today. Lead Tomorrow.

2 ACTEX SOA Eam MLC Sudy Manual Fall 7 Ediion Johnny Li, P.h.D., FSA Andrew Ng, Ph.D., FSA ACTEX Learning New Harford, Connecicu

3 ACTEX Learning Learn Today. Lead Tomorrow. Acuarial & Financial Risk Resource Maerials Since 97 Copyrigh 7 SRBooks, Inc. ISBN: Prined in he Unied Saes of America. No porion of his ACTEX Sudy Manual may be reproduced or ransmied in any par or by any means wihou he permission of he publisher.

4 Preface P- Conens Preface P-7 Syllabus Reference P- Flow Char P-3 Chaper Some Facual Informaion C-. Tradiional Life Insurance Conracs C-. Modern Life Insurance Conracs C-3.3 Underwriing C-3.4 Life Annuiies C-4.5 Pensions C-6 Chaper Survival Disribuions C-. Age-a-deah Random Variables C-. Fuure Lifeime Random Variable C-4.3 Acuarial Noaion C-6.4 Curae Fuure Lifeime Random Variable C-.5 Force of Moraliy C- Eercise C- Soluions o Eercise C-7 Chaper Life Tables C-. Life Table Funcions C-. Fracional Age Assumpions C-6.3 Selec-and-Ulimae Tables C-8.4 Momens of Fuure Lifeime Random Variables C-9.5 Useful Shorcus C-39 Eercise C-43 Soluions o Eercise C-5 Chaper 3 Life Insurances C3-3. Coninuous Life Insurances C3-3. Discree Life Insurances C mhly Life Insurances C Relaing Differen Policies C Recursions C Relaing Coninuous, Discree and mhly Insurance C Useful Shorcus C3-46 Eercise 3 C3-48 Soluions o Eercise 3 C3-6 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

5 P- Preface Chaper 4 Life Annuiies C4-4. Coninuous Life Annuiies C4-4. Discree Life Annuiies (Due) C Discree Life Annuiies (Immediae) C mhly Life Annuiies C Relaing Differen Policies C Recursions C Relaing Coninuous, Discree and mhly Life Annuiies C Useful Shorcus C4-44 Eercise 4 C4-47 Soluions o Eercise 4 C4-6 Chaper 5 Premium Calculaion C5-5. Tradiional Insurance Policies C5-5. Ne Premium and Equivalence Principle C Ne Premiums for Special Policies C5-5.4 The Loss-a-issue Random Variable C Percenile Premium and Profi C5-7 Eercise 5 C5-38 Soluions o Eercise 5 C5-55 Chaper 6 Ne Premium Reserves C6-6. The Prospecive Approach C6-6. The Recursive Approach: Basic Idea C The Recursive Approach: Furher Applicaions C The Rerospecive Approach C6-33 Eercise 6 C6-4 Soluions o Eercise 6 C6-65 Chaper 7 Insurance Models Including Epenses C7-7. Gross Premium C7-7. Gross Premium Reserve C Epense Reserve and Modified Reserve C Basis, Asse Share and Profi C7-3 Eercise 7 C7-39 Soluions o Eercise 7 C7-57 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

6 Preface P-3 Chaper 8 Muliple Decremen Models: Theory C8-8. Muliple Decremen Table C8-8. Forces of Decremen C Associaed Single Decremen C8-8.4 Discree Jumps C8-3 Eercise 8 C8-9 Soluions o Eercise 8 C8-4 Chaper 9 Muliple Decremen Models: Applicaions C9-9. Calculaing Acuarial Presen Values of Cash Flows C9-9. Calculaing Reserve and Profi C Cash Values C Calculaing Asse Shares under Muliple Decremen C9-3 Eercise 9 C9-8 Soluions o Eercise 9 C9-4 Chaper Muliple Sae Models C-. Discree-ime Markov Chain C-4. Coninuous-ime Markov Chain C-3.3 Kolmogorov s Forward Equaions C-9.4 Calculaing Acuarial Presen Value of Cash Flows C-3.5 Calculaing Reserves C-39 Eercise C-44 Soluions o Eercise C-63 Chaper Muliple Life Funcions C-. Muliple Life Sauses C-. Insurances and Annuiies C-7.3 Dependen Life Models C-3 Eercise C-43 Soluions o Eercise C-63 Chaper Ineres Rae Risk C-. Yield Curves C-. Ineres Rae Scenario Models C-3.3 Diversifiable and Non-Diversifiable Risks C-7 Eercise C-6 Soluions o Eercise C-37 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

7 P-4 Preface Chaper 3 Profi Tesing C3-3. Profi Vecor and Profi Signaure C3-3. Profi Measures C3-3.3 Using Profi Tes o Compue Premiums and Reserves C3-6 Eercise 3 C3-4 Soluions o Eercise 3 C3-3 Chaper 4 Universal Life Insurance C4-4. Basic Policy Design C4-4. Cos of Insurance and Surrender Value C Oher Policy Feaures C Projecing Accoun Values C4-4.5 Profi Tesing C Asse Shares for Universal Life Policies C4-4 Eercise 4 C4-43 Soluions o Eercise 4 C4-53 Chaper 5 Paricipaing Insurance C5-5. Dividends C5-5. Bonuses C5- Eercise 5 C5-4 Soluions o Eercise 5 C5-8 Chaper 6 Pension Mahemaics C6-6. The Salary Scale Funcion C6-6. Pension Plans C6-6.3 Seing he DC Conribuion Rae C DB Plans and Service Table C Funding of DB Plans C6-39 Eercise 6 C6-45 Soluions o Eercise 6 C6-6 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

8 Preface P-5 Appendi Numerical Techniques A-. Numerical Inegraion A-. Euler s Mehod A-7.3 Solving Sysems of ODEs wih Euler s Mehod A- Appendi Review of Probabiliy A-. Probabiliy Laws A-. Random Variables and Epecaions A-.3 Special Univariae Probabiliy Disribuions A-6.4 Join Disribuion A-9.5 Condiional and Double Epecaion A-.6 The Cenral Limi Theorem A- Eam MLC: General Informaion T- Mock Tes T- Soluion T-9 Mock Tes T- Soluion T-8 Mock Tes 3 T3- Soluion T3-9 Mock Tes 4 T4- Soluion T4-9 Mock Tes 5 T5- Soluion T5-9 Mock Tes 6 T6- Soluion T6-9 Mock Tes 7 T7- Soluion T7-9 Mock Tes 8 T8- Soluion T8-8 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

9 P-6 Preface Suggesed Soluions o MLC May S- Suggesed Soluions o MLC Nov S-7 Suggesed Soluions o MLC May 3 S-9 Suggesed Soluions o MLC Nov 3 S-45 Suggesed Soluions o MLC April 4 S-55 Suggesed Soluions o MLC Oc 4 S-69 Suggesed Soluions o MLC April 5 S-8 Suggesed Soluions o MLC Oc 5 S-97 Suggesed Soluions o MLC May 6 S-9 Suggesed Soluions o MLC Oc 6 S-3 Suggesed Soluions o MLC April 7 S-33 Suggesed Soluions o Sample Srucural Quesions S-45 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

10 Preface P-7 Preface Thank you for choosing ACTEX. A new version of Eam MLC is launched in Spring 4. The new Eam MLC is significanly differen from he old one, mos noably in he following aspecs: () Wrien-answer quesions are inroduced and form a major par of he eaminaion. () The number of official ebooks is reduced from wo o one. The new official ebook, Acuarial Mahemaics for Life Coningen Risks nd ediion (AMLCR), conains a lo more echnical maerials han oher ebooks wrien on he same opic. (3) The level of cogniive skills demanded from candidaes is much higher. In paricular, he new learning objecives require candidaes o no only calculae numerical values bu also, for eample, inerpre he resuls hey obain. (4) Several new (and more advanced) opics, such as paricipaing insurance, are added o he syllabus. Because of hese major changes, ACTEX have decided o bring you his new sudy manual, which is wrien o fi he new eam. We know very well ha you may be worried abou wrien-answer quesions. To help you bes prepare for he new eam, his manual conains some 5 wrien-answer quesions for you o pracice. Eigh full-lengh mock eams, wrien in eacly he same forma as ha announced in SoA s Eam MLC Inroducory Noe, are also provided. Many of he wrien-answer quesions in our mock eams are highly challenging! We are sorry for giving you a hard ime, bu we do wan you o succeed in he real eam. The learning oucomes of he new eam syllabus require candidaes o be able o inerpre a lo of acuarial conceps. This skill is drilled eensively in our wrien-answer pracice problems, which ofen ask you o inerpre a cerain acuarial formula or o eplain your calculaion. Also, as seen in SoA s Eam MLC Sample Wrien-Answer Quesions (e.g., #9), you may be asked in he new eam o define or describe a cerain insurance produc or acuarial erminology. To help you prepare for his ype of eam problems, we have prepared a special chaper (Chaper ), which conains definiions and descripions of various producs and erminologies. The special chaper is wrien in a fac shee syle so ha you can remember he key poins more easily. Proofs and derivaions are anoher key challenge. In he new eam, you are highly likely o be asked o prove or derive somehing. This is evidenced by, for eample, problem #4 in SoA s Eam MLC Sample Wrien-Answer Quesions, which demands a mahemaical derivaion of he Kolmogorov forward differenial equaions for a cerain ransiion probabiliy. In his new sudy manual, we do each (and drill) you how o prove or derive imporan formulas. This is in sark conras o some oher eam prep producs in which proofs and derivaions are downplayed, if no omied. We have paid special aenion o he opics ha are newly inroduced in he recen wo syllabus updaes. Seven full-lengh chapers (Chapers,, 6) and wo secions (amoun o more Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

11 P-8 Preface han 3 pages) are especially devoed o hese opics. Moreover, insead of reaing he new opics as orphans, we demonsrae, as far as possible, how hey can be relaed o he old opics in an eam seing. This is very imporan for you, because muliple learning oucomes can be eamined in one single eam quesion. We have made our bes effor o ensure ha all opics in he syllabus are eplained and praciced in sufficien deph. For your reference, a deailed mapping beween his sudy manual and he official ebook is provided on pages P- o P-. Besides he opics specified in he eam syllabus, you also need o know a range of numerical echniques in order o succeed. These echniques include, for eample, Euler s mehod, which is involved in SoA s Eam MLC Sample Muliple-Choice Quesion #99. We know ha quie a few of you have no even heard of Euler s mehod before, so we have prepared a special chaper (Appendi, appended o he end of he sudy manual) o each you all numerical echniques required for his eam. In addiion, whenever a numerical echnique is used, we clearly poin ou which echnique i is, leing you follow our eamples and eercises more easily. Oher disinguishing feaures of his sudy manual include: We use graphics eensively. Graphical illusraions are probably he mos effecive way o eplain formulas involved in Eam MLC. The eensive use of graphics can also help you remember various conceps and equaions. A sleek layou is used. The fon size and spacing are chosen o le you feel more comforable in reading. Imporan equaions are displayed in eye-caching boes. Raher han spliing he manual ino iny unis, each of which ells you a couple of formulas only, we have carefully grouped he eam opics ino 7 chapers. Such a grouping allows you o more easily idenify he linkages beween differen conceps, which, as we menioned earlier, are essenial for your success. Insead of giving you a long lis of formulas, we poin ou which formulas are he mos imporan. Having read his sudy manual, you will be able o idenify he formulas you mus remember and he formulas ha are jus varians of he key ones. We do no wan o overwhelm you wih verbose eplanaions. Whenever possible, conceps and echniques are demonsraed wih eamples and inegraed ino he pracice problems. We wrie he pracice problems and he mock eams in a similar forma as he released eam and sample quesions. This will enable you o comprehend quesions more quickly in he real eam. On page P-3, you will find a flow char showing how differen chapers of his manual are conneced o one anoher. You should firs sudy Chapers o in order. Chaper will give you some background facual informaion; Chapers o 4 will build you a solid foundaion; and Chapers 5 o will ge you o he core of he eam. You should hen sudy Chapers o 6 in any order you wish. Immediaely afer reading a chaper, do all pracice problems we provide for ha chaper. Make sure ha you undersand every single pracice problem. Finally, work on he mock eams. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

12 Preface P-9 Before you begin your sudy, please download he eam syllabus from SoA s websie: hps:// On he las page of he eam syllabus, you will find a link o Eam MLC Tables, which are frequenly used in he eam. You should keep a copy of he ables, as we are going o refer o hem from ime o ime. You should also check he eam home page periodically for updaes, correcions or noices. If you find a possible error in his manual, please le us know a he Cusomer Feedback link on he ACTEX homepage ( Any confirmed erraa will be posed on he ACTEX websie under he Erraa & Updaes link. Enjoy your sudy! Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

13 P- Preface Syllabus Reference Our Manual AMLCR Chaper : Some Facual Informaion..6 Chaper : Survival Disribuions.., Chaper : Life Tables. 3., , 3.8, ,.6.5 Chaper 3: Life Insurances , 4.4.5, 4.4.7, , 4.4.6, 4.4.7, , Chaper 4: Life Annuiies , 5.4., 5.9, , , Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

14 Preface P- Our Manual AMLCR Chaper 5: Premium Calculaion 5. 6., Chaper 6: Ne Premium Reserves 6. 7., 7.3., Chaper 7: Insurance Models Including Epenses , , 7.3.4, Chaper 8: Muliple Decremen Models: Theory , , Chaper 9: Muliple Decremen Models: Applicaions Chaper : Muliple Sae Models , 8.3, , Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

15 P- Preface Our Manual AMLCR Chaper : Muliple Life Funcions Chaper : Ineres Rae Risk ,.4 Chaper 3: Profi Tesing ,.7 Chaper 4: Universal Life Insurance , 3.4., 3.4., , 3.4., 3.4.5, , Chaper 5: Paricipaing Insurance Chaper 6: Pension Mahemaics , , Appendi : Numerical Techniques A. 8.6 A A Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

16 Preface P-3 Flow Char. Survival Disribuions. Some Facual Informaion. Life Tables 5. Premium Calculaion. Ineres Rae Risk 3. Life Insurances 6. Ne Premium Reserves 4. Life Annuiies 7. Insurance Models Including Epenses A. Numerical Techniques 8. Muliple Decremen Models: Theory 9. Muliple Decremen Models: Applicaions 6. Pension Mahemaics. Muliple Sae Models. Muliple Life Funcions 5. Paricipaing Insurance 3. Profi Tesing 4. Universal Life Insurance Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

17 P-4 Preface Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

18 Chaper : Some Facual Informaion C- Chaper Some Facual Informaion This chaper serves as a summary of Chaper in AMLCR. I conains descripions of various life insurance producs and pension plans. There is absoluely no mahemaics in his chaper. You should know (and remember) he informaion presened in his chaper, because in he wrien answer quesions, you may be asked o define or describe a cerain pension plan or life insurance policy. Mos of he maerials in his chaper are presened in a fac shee syle so ha you can remember he key poins more easily. Many of he policies and plans menioned in his chaper will be discussed in deail in laer pars of his sudy guide.. Tradiional Life Insurance Conracs Whole life insurance A whole life insurance pays a benefi on he deah of he policyholder whenever i occurs. The following diagram illusraes a whole life insurance sold o a person age. A benefi (he sum insured) is paid here (Age ) Deah occurs Time from now The amoun of benefi is ofen referred o as he sum insured. The policyholder, of course, has o pay he price of policy. In insurance cone, he price of a policy is called he premium, which may be payable a he beginning of he policy, or periodically hroughou he life ime of he policy. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

19 C- Chaper : Survival Disribuions Term life insurance A erm life insurance pays a benefi on he deah of he policyholder, provided ha deah occurs before he end of a specified erm. Deah occurs here: Pay a benefi Deah occurs here: Pay nohing n (Age ) Time from now The ime poin n in he diagram is called he erm or he mauriy dae of he policy. Endowmen insurance An endowmen insurance offers a benefi paid eiher on he deah of he policyholder or a he end of a specified erm, whichever occurs earlier. Deah occurs here: Pay he sum insured on deah n (Age ) Pay he sum insured a ime n if he policyholder is alive a ime n Time from now These hree ypes of radiional life insurance will be discussed in Chaper 3 of his sudy guide. Paricipaing (wih profi) insurance Any premium colleced from he policyholder will be invesed, for eample, in he bond marke. In a paricipaing insurance, he profis earned on he invesed premiums are shared wih he policyholder. The profi share can ake differen forms, for eample, cash dividends, reduced premiums or increased sum insured. This produc ype will be discussed in deail in Chaper 5 of his sudy manual. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

20 Chaper : Some Facual Informaion C-3. Modern Life Insurance Conracs Modern life insurance producs are usually more fleible and ofen involve an invesmen componen. The able below summarizes he feaures of several modern life insurance producs. Produc Universal life insurance Uniized wih-profi insurance Feaures Combines invesmen and life insurance Premiums are fleible, as long as he accumulaed value of he premiums is enough o cover he cos of insurance Similar o radiional paricipaing insurance Premiums are used o purchase shares of an invesmen fund. The income from he invesmen fund increases he sum insured. The benefi is linked o he performance of an invesmen fund. Equiy-linked insurance Eamples: equiy-indeed annuiies (EIA), uni-linked policies, segregaed fund policies, variable annuiy conracs Usually, invesmen guaranees are provided. In Chaper 4 of his sudy guide, we will discuss universal life insurance policies in deail.. 3 Underwriing Underwriing refers o he process of collecing and evaluaing informaion such as age, gender, smoking habis, occupaion and healh hisory. The purposes of his process are: To classify poenial policyholders ino broadly homogeneous risk caegories To deermine if addiional premium has o be charged. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

21 C-4 Chaper : Survival Disribuions The following able summarizes a ypical caegorizaion of poenial policyholders. Caegory Preferred lives Normal lives Raed lives Uninsurable lives Characerisics Have very low moraliy risk Have some risk bu no addiional premium has o be charged Have more risk and addiional premium has o be charged Have oo much risk and herefore no insurable Underwriing is an imporan process, because wih no (or insufficien) underwriing, here is a risk of adverse selecion; ha is, he insurance producs end o arac high risk individuals, leading o ecessive claims. In Chaper, we will inroduce he selec-and-ulimae able, which is closely relaed o underwriing.. 4 Life Annuiies A life annuiy is a benefi in he form of a regular series of paymens, condiional on he survival of he policyholder. There are differen ypes of life annuiies. Single premium immediae annuiy (SPIA) The annuiy benefi of a SPIA commences as soon as he conrac is wrien. The policyholder pays a single premium a he beginning of he conrac. Annuiy benefis are paid (Age ) Deah occurs Time from now A single premium is paid a he beginning of he conrac Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

22 Chaper : Some Facual Informaion C-5 Single premium deferred annuiy (SPDA) The annuiy benefi of a SPDA commences a some fuure specified dae (say n years from now). The policyholder pays a single premium a he beginning of he conrac. The annuiy benefi begins a ime n (Age ) n Deah occurs Time from now A single premium is paid a he beginning of he conrac Regular Premium Deferred Annuiy (RPDA) An RPDA is idenical o a SPDA ecep ha he premiums are paid periodically over he deferred period (i.e., before ime n). These hree annuiy ypes will be discussed in greaer deph in Chaper 4 of his sudy guide. Some life annuiies are issued o wo lives (a husband and wife). These life annuiies can be classified as follows. Join life annuiy Las survivor annuiy Reversionary annuiy The annuiy benefi ceases on he firs deah of he couple. The annuiy benefi ceases on he second deah of he couple. The annuiy benefi begins on he firs deah of he couple, and ceases on he second deah. These annuiies will be discussed in deail in Chaper of his sudy guide. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

23 C-6 Chaper : Survival Disribuions. 5 Pensions A pension provides a lump sum and/or annuiy benefi upon an employee s reiremen. In he following able, we summarize a ypical classificaion of pension plans: Defined conribuion (DC) plans Defined benefi (DB) plans The reiremen benefi from a DC plan depends on he accumulaion of he deposis made by he employ and employee over he employee s working life ime. The reiremen benefi from a DB plan depends on he employee s service and salary. Final salary plan: he benefi is a funcion of he employee s final salary. Career average plan: he benefi is a funcion of he average salary over he employee s enire career in he company. Pension plans will be discussed in deail in Chaper 6 of his sudy guide. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

24 Chaper : Survival Disribuions C- Chaper Survival Disribuions OBJECTIVES. To define fuure lifeime random variables. To specify survival funcions for fuure lifeime random variables 3. To define acuarial symbols for deah and survival probabiliies and develop relaionships beween hem 4. To define he force of moraliy In Eam FM, you valued cash flows ha are paid a some known fuure imes. In Eam MLC, by conras, you are going o value cash flows ha are paid a some unknown fuure imes. Specifically, he imings of he cash flows are dependen on he fuure lifeime of he underlying individual. These cash flows are called life coningen cash flows, and he sudy of hese cash flows is called life coningencies. I is obvious ha an imporan par of life coningencies is he modeling of fuure lifeimes. In his chaper, we are going o sudy how we can model fuure lifeimes as random variables. A few simple probabiliy conceps you learn in Eam P will be used.. Age-a-deah Random Variable Le us begin wih he age-a-deah random variable, which is denoed by T. The definiion of T can be easily seen from he diagram below. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

25 C- Chaper : Survival Disribuions Deah occurs T Age The age-a-deah random variable can ake any value wihin [, ). Someimes, we assume ha no individual can live beyond a cerain very high age. We call ha age he limiing age, and denoe i by ω. If a limiing age is assumed, hen T can only ake a value wihin [, ω]. We regard T as a coninuous random variable, because i can, in principle, ake any value on he inerval [, ) if here is no limiing age or [, ω] if a limiing age is assumed. Of course, o model T, we need a probabiliy disribuion. The following noaion is used hroughou his sudy guide (and in he eaminaion). F () Pr(T ) is he (cumulaive) disribuion funcion of T. d f () () d F is he probabiliy densiy funcion of T. For a small inerval Δ, he produc f ()Δ is he (approimae) probabiliy ha he age a deah is in beween and Δ. In life coningencies, we ofen need o calculae he probabiliy ha an individual will survive o a cerain age. This moivaes us o define he survival funcion: S () Pr(T > ) F (). Noe ha he subscrip indicaes ha hese funcions are specified for he age-a-deah random variable (or equivalenly, he fuure lifeime of a person age now). No all funcions can be regarded as survival funcions. A survival funcion mus saisfy he following requiremens:. S (). This means every individual can live a leas years.. S (ω) or lim S ( ). This means ha every individual mus die evenually. 3. S () is monoonically decreasing. This means ha, for eample, he probabiliy of surviving o age 8 canno be greaer han ha of surviving o age 7. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

26 Chaper : Survival Disribuions C-3 Summing up, f (), F () and S () are relaed o one anoher as follows. F O R M U L A Relaions beween f (), F () and S () f d d () () () d F d S, (.) S ( ) f ( u)du f ( u)du F ( ), (.) b Pr(a < T b) f ( u)du F ( b) F ( a) S ( a) S ( b). (.3) a Noe ha because T is a coninuous random variable, Pr(T c) for any consan c. Now, le us consider he following eample. Eample. [Srucural Quesion] You are given ha S () / for. (a) Verify ha S () is a valid survival funcion. (b) Find epressions for F () and f (). (c) Calculae he probabiliy ha T is greaer han 3 and smaller han 6. Soluion (a) Firs, we have S () /. Second, we have S () /. Third, he firs derivaive of S () is /, indicaing ha S () is non-increasing. Hence, S () is a valid survival funcion. (b) We have F () S () /, for. Also, we have and f () d d F () /, for. (c) Pr(3 < T < 6) S (3) S (6) ( 3/) ( 6/).3. [ END ] Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

27 C-4 Chaper : Survival Disribuions. Fuure Lifeime Random Variable Consider an individual who is age now. Throughou his e, we use () o represen such an individual. Insead of he enire lifeime of (), we are ofen more ineresed in he fuure lifeime of (). We use T o denoe he fuure lifeime random variable for (). The definiion of T can be easily seen from he diagram below. T Age Now Deah occurs T Time from now [Noe: For breviy, we may only display he porion saring from age (i.e., ime ) in fuure illusraions.] If here is no limiing age, T can ake any value wihin [, ). If a limiing age is assumed, hen T can only ake a value wihin [, ω ]. We have o subrac because he individual has aained age a ime already. We le S () be he survival funcion for he fuure lifeime random variable. The subscrip here indicaes ha he survival funcion is defined for a life who is age now. I is imporan o undersand ha when modeling he fuure lifeime of (), we always know ha he individual is alive a age. Thus, we may evaluae S () as a condiional probabiliy: S ( ) Pr( T > ) Pr( T > T > ) Pr( T > T > ) Pr( T > ) S( ). Pr( T > ) Pr( T > ) S ( ) The hird sep above follows from he equaion Eam P. Pr( A B) Pr( A B), which you learn in Pr( B) Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

28 Chaper : Survival Disribuions C-5 Survival Funcion for he Fuure Lifeime Random Variable S () S ( ) S ( ) F O R M U L A (.4) Wih S (), we can obain F () and f () by using respecively. F () S () and f () d F (), d Eample. [Srucural Quesion] You are given ha S () / for. (a) Find epressions for S (), F () and f (). (b) Calculae he probabiliy ha an individual age now can survive o age 5. (c) Calculae he probabiliy ha an individual age now will die wihin 5 years. Soluion (a) In his par, we are asked o calculae funcions for an individual age now (i.e., ). Here, ω and herefore hese funcions are defined for 9 only. S( ) ( ) / Firs, we have S(), for 9. S () / 9 Second, we have F () S () /9, for 9. Finally, we have f d () F (). d 9 (b) The probabiliy ha an individual age now can survive o age 5 is given by 5 5 Pr(T > 5) S (5). 9 6 (c) The probabiliy ha an individual age now will die wihin 5 years is given by Pr(T 5) F (5) S (5) 6. [ END ] Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

29 C-6 Chaper : Survival Disribuions. 3 Acuarial Noaion For convenience, we have designaed acuarial noaion for various ypes of deah and survival probabiliies. Noaion : p We use p o denoe he probabiliy ha a life age now survives o years from now. By definiion, we have p Pr(T > ) S (). When, we can omi he subscrip on he lef-hand-side; ha is, we wrie p as p. Noaion : q We use q o denoe he probabiliy ha a life age now dies before aaining age. By definiion, we have q Pr(T ) F (). When, we can omi he subscrip on he lef-hand-side; ha is, we wrie q as q. Noaion 3: u q We use u q o denoe he probabiliy ha a life age now dies beween ages and u. By definiion, we have u q Pr( < T u) F ( u) F () S () S ( u). When u, we can omi he subscrip u; ha is, we wrie q as q. Noe ha when we describe survival disribuions, p always means a survival probabiliy, while q always means a deah probabiliy. The beween and u means ha he deah probabiliy is deferred by years. We read u as deferred u. I is imporan o remember he meanings of hese hree acuarial symbols. Le us sudy he following eample. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

30 Chaper : Survival Disribuions C-7 Eample.3 Epress he probabiliies associaed wih he following evens in acuarial noaion. (a) A new born infan dies no laer han age 45. (b) A person age now survives o age 38. (c) A person age 57 now survives o age 6 bu dies before aaining age 65. Assuming ha S () e.5 for, evaluae he probabiliies. Soluion (a) The probabiliy ha a new born infan dies no laer han age 45 can be epressed as 45 q. [Here we have q for a deah probabiliy, and 45.] Furher, 45 q F (45) S (45).43. (b) The probabiliy ha a person age now survives o age 38 can be epressed as 8 p. [Here we have p for a survival probabiliy, and 38 8.] S(38) Furher, we have 8 p S (8) S () (c) The probabiliy ha a person age 57 now survives o age 6 bu dies before aaining age 65 can be epressed as 3 5 q 57. [Here, we have q for a (deferred) deah probabiliy, 57, , and u ] S(6) Furher, we have 3 5 q 57 S 57 (3) S 57 (8) S (57) S (65) S (57) [ END ] Oher han heir meanings, you also need o know how hese symbols are relaed o one anoher. Here are four equaions ha you will find very useful. Equaion : p q This equaion arises from he fac ha here are only wo possible oucomes: dying wihin years or surviving o years from now. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

31 C-8 Chaper : Survival Disribuions Equaion : u p p u p The meaning of his equaion can be seen from he following diagram. Survive from ime o : probabiliy p Survive from ime o u: probabiliy u p u (Age ) (Age ) Time from now Survive from ime o u: probabiliy u p Mahemaically, we can prove his equaion as follows: S ( u ) S ( ) S ( u ) p S u S S u p p u ( ) ( ) ( ) u S( ) S( ) S( ). Equaion 3: u q u q q p u p This equaion arises naurally from he definiion of u q. We have u q Pr( < T u) F ( u) F ( ) u q u q q. Also, u q Pr( < T u) S () S ( u) p u p. Equaion 4: u q p u q The reasoning behind his equaion can be undersood from he following diagram: Survive from ime o ime : probabiliy p Deah occurs: prob. u q u (Age ) (Age ) Time from now u q Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

32 Chaper : Survival Disribuions C-9 Mahemaically, we can prove his equaion as follows: u q p u p (from Equaion 3) p p u p (from Equaion ) p ( u p ) p u q (from Equaion ) Here is a summary of he equaions ha we jus inroduced. F O R M U L A Relaions beween p, q and u q p q, (.5) u p p u p, (.6) u q u q q p u p p u q. (.7) Le us go hrough he following eample o see how hese equaions are applied. Eample.4 You are given: (i) p.99 (ii) p.985 (iii) 3p.95 (iv) q 3. Calculae he following: (a) p 3 (b) p (c) p (d) 3 p (e) q Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

33 C- Chaper : Survival Disribuions Soluion (a) p 3 q (b) p p p (c) Consider 3 p p p 3.95 p.98 p.9694 (d) 3 p p p (e) q p q p ( p ).99 (.9694).33 [ END ]. 4 Curae Fuure Lifeime Random Variable In pracice, acuaries use Ecel eensively, so a discree version of he fuure lifeime random variable would be easier o work wih. We define K T, where y means he inegral par of y. For eample, call K he curae fuure lifeime random variable., and.99. We I is obvious ha K is a discree random variable, since i can only ake non-negaive inegral values (i.e.,,,, ). The probabiliy mass funcion for K can be derived as follows: Pr(K ) Pr( T < ) q, Pr(K ) Pr( T < ) q, Pr(K ) Pr( T < 3) q, Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

34 Chaper : Survival Disribuions C- Inducively, we have F O R M U L A Probabiliy Mass Funcion for K Pr(K k) k q, k,,, (.8) The cumulaive disribuion funcion can be derived as follows: Pr(K k) Pr(T < k ) k q, for k,,,. I is jus ha simple! Now, le us sudy he following eample, which is aken from a previous SoA Eam. Eample.5 [Course 3 Fall 3 #8] For (): (i) K is he curae fuure lifeime random variable. (ii) q k.(k ), k,,,, 9 Calculae Var(K 3). (A). (B). (C).3 (D).4 (E).5 Soluion The noaion means minimum. So here K 3 means min(k, 3). For convenience, we le W min(k, 3). Our job is o calculae Var(W). Noe ha he only possible values ha W can ake are,,, and 3. To accomplish our goal, we need he probabiliy funcion of W, which is relaed o ha of K. The probabiliy funcion of W is derived as follows: Pr(W ) Pr(K ) q. Pr(W ) Pr(K ) q p q Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

35 C- Chaper : Survival Disribuions ( q )q (.)..8 Pr(W ) Pr(K ) q p q p p q ( q )( q ) q Pr(W 3) Pr(K 3) Pr(K ) Pr(K ) Pr(K ).54. From he probabiliy funcion for W, we obain E(W) and E(W ) as follows: E(W) E(W ) This gives Var(W) E(W ) [E(W)] Hence, he answer is (A). [ END ]. 5 Force of Moraliy In Eam FM, you learn a concep called he force of ineres, which measures he amoun of ineres credied in a very small ime inerval. By using his concep, you valued, for eample, annuiies ha make payous coninuously. In his eam, you will encouner coninuous life coningen cash flows. To value such cash flows, you need a funcion ha measures he probabiliy of deah over a very small ime inerval. This funcion is called he force of moraliy. Consider an individual age now. The force of moraliy for his individual years from now is denoed by μ or μ (). A ime, he (approimae) probabiliy ha his individual dies wihin a very small period of ime Δ is μ Δ. The definiion of μ can be seen from he following diagram. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

36 Chaper : Survival Disribuions C-3 Survive from ime o ime : Prob. S () Deah occurs during o Δ: Prob. μ Δ Δ Time from now Deah beween ime and Δ: Prob. (measured a ime ) f ()Δ From he diagram, we can also ell ha f () Δ S ()μ Δ. I follows ha f () S ()μ p μ. This is an eremely imporan relaion, which will be used hroughou his sudy manual. Recall ha f ( ) F ( ) S ( ). This yields he following equaion: S ( ) μ, S ( ) which allows us o find he force of moraliy when he survival funcion is known. Recall ha dln, and ha by chain rule, d We can rewrie he previous equaion as follows: Replacing by u, μ μ μ μ d[ln S ( )] μ d μ d d[ln S ( )]. u u u du d ln g ( ) g ( ) for a real-valued funcion g. d g( ) S ( ) S ( ) du ln S du d[ln S d[ln S S ( ) ep ( u)] ( u)] ( ) ln S μ u () du. This allows us o find he survival funcion when he force of moraliy is known. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

37 C-4 Chaper : Survival Disribuions F O R M U L A Relaions beween μ, f () and S () f () S ()μ p μ, (.9) S ( ) μ, (.) S ( ) S ( ) ep d. μ u u (.) No all funcions can be used for he force of moraliy. We require he force of moraliy o saisfy he following wo crieria: (i) μ for all and. (ii) μ du u. Crierion (i) follows from he fac ha μ Δ is a measure of probabiliy, while Crierion (ii) follows from he fac ha lim ( ). S Noe ha he subscrip indicaes he age a which deah occurs. So you may use μ o denoe he force of moraliy a age. For eample, μ refers o he force of moraliy a age. The wo crieria above can hen be wrien alernaively as follows: (i) μ for all. (ii) μ d. The following wo specificaions of he force of moraliy are ofen used in pracice. Gomperz law μ Bc Makeham s law μ A Bc In he above, A, B and c are consans such ha A B, B > and c >. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

38 Chaper : Survival Disribuions C-5 Le us sudy a few eamples now. Eample.6 [Srucural Quesion] For a life age now, you are given: (a) Find μ. (b) Find f (). ( ) S (), <. Soluion ( ) S ( ) (a) μ. S ( ) ( ) (b) You may work direcly from S (), bu since we have found μ already, i would be quicker o find f () as follows: f () S ()μ. 5 ( ) [ END ] Eample.7 [Srucural Quesion] For a life age now, you are given μ.,. (a) Is μ a valid funcion for he force of moraliy of ()? (b) Find S (). (c) Find f (). Soluion (a) Firs, i is obvious ha μ for all and. Second, μ d u. u d u. u u. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

39 C-6 Chaper : Survival Disribuions Hence, i is a valid funcion for he force of moraliy of (). (b) S () ep udu ep.udu ep(. ) μ. (c) f () S ()μ.ep(. ). [ END ] Eample.8 [Course 3 Fall #35] You are given: (i) R ep μ d (ii) S ep ( μ k)d (iii) k is a consan such ha S.75R. Deermine an epression for k. (A) ln(( q )/ (.75q )) (B) ln((.75q ) / ( p )) (C) ln((.75p ) / ( p )) (D) ln(( p )/ (.75q )) (E) ln((.75q ) / ( q )) Soluion Firs, R S () p q. Second, S ep Since S.75R, we have k k k ( μ k)du e ep μ du e S() e p. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

40 Chaper : Survival Disribuions C-7 Hence, he answer is (A). k e p.75q e k e k.75q p p.75q. p q k ln ln.75q.75q [ END ] Eample.9 [Srucural Quesion] (a) Show ha when μ Bc, we have where g is a consan ha you should idenify. c ( c ) p g, (b) For a moraliy able consruced using he above force of moraliy, you are given ha p and p Calculae he values of B and c. Soluion (a) To prove he equaion, we should make use of he relaionship beween he force of moraliy and p. s B p ep μ sds ep Bc ds ep c ( c ). ln c This gives g ep( B/lnc). (b) From (a), we have g c 5 ( c ) and c g 5 ( c ). This gives c c ln(.78743) ln(.8676) Solving his equaion, we obain c.. Subsiuing back, we obain g and B.5.. [ END ] Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

41 C-8 Chaper : Survival Disribuions Now, le us sudy a longer srucural quesion ha inegraes differen conceps in his chaper. Eample. [Srucural Quesion] The funcion 8 8 has been proposed for he survival funcion for a moraliy model. (a) Sae he implied limiing age ω. (b) Verify ha he funcion saisfies he condiions for he survival funcion S (). (c) Calculae p. (d) Calculae he survival funcion for a life age. (e) Calculae he probabiliy ha a life aged will die beween ages 3 and 4. (f) Calculae he force of moraliy a age 5. Soluion (a) Since 8 ω ω S ( ω), 8 We have ω ω 8 (ω 9)(ω ) ω 9 or ω (rejeced). Hence, he implied limiing age is 9. (b) We need o check he following hree condiions: 8 (i) S () 8 8 ω ω (ii) S ( ω) 8 d (iii) S ( ) < d 8 Therefore, he funcion saisfies he condiions for he survival funcion S (). Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

42 Chaper : Survival Disribuions C-9 8 (c) p S() (d) (d) S (9 )( ) S( ) ( ) 8 S () (9 )( ) 8 (7 )( ) (e) The required probabiliy is q p p (7 )( ) (7 )( ) (f) Firs, we find an epression for μ. Hence, μ 5 S ( ) 8 μ. S ( ) (9 )( ) (9 )( ) (9 5)(5 ) [ END ] You may be asked o prove some formulas in he srucural quesions of Eam MLC. Please sudy he following eample, which involves several proofs. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

43 C- Chaper : Survival Disribuions Eample. [Srucural Quesion] Prove he following equaions: d (a) p p μ d (b) q p μ ds s s ω (c) p μ d Soluion d d d (a) LHS p ep( sds) ep( sds) sds p ( ) d d μ μ d μ μ RHS (b) LHS q Pr(T ) s s f ( s)ds p μ ds RHS (c) LHS ω p μ d ω f ( ) d ω q RHS [ END ] Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

44 Chaper : Survival Disribuions C- Eercise. [Srucural Quesion] You are given: S(),. (a) Find F (). (b) Find f (). (c) Find S (). (d) Calculae p. (e) Calculae 5 q 3.. You are given: Find an epression for p You are given: Find μ. (3 ) f(), 9 for < 3 f(), <. 4. [Srucural Quesion] You are given: μ, <. (a) Find S () for < 8. (b) Compue 4 p. (c) Find f () for < You are given: μ, for <. Find he probabiliy ha he age a deah is in beween and You are given: α (i) S () ω < ω, α >. (ii) μ 4 μ. Find ω. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

45 C- Chaper : Survival Disribuions 7. Epress he probabiliies associaed wih he following evens in acuarial noaion. (a) A new born infan dies no laer han age 35. (b) A person age now survives o age 5. (c) A person age 4 now survives o age 5 bu dies before aaining age 55. Assuming ha S () e.5 for, evaluae he probabiliies. 8. You are given: () S, <. Find he probabiliy ha a person aged will die beween he ages of 5 and You are given: (i) p.98 (ii) p.985 (iii) 5q.775 Calculae he following: (a) 3 p (b) p 3 (c) 3 q. You are given: q k.(k ), k,,,, 9. Calculae he following: (a) Pr(K ) (b) Pr(K ). [Srucural Quesion] You are given μ μ for all. (a) Find an epression for Pr(K k), for k,,,, in erms of μ and k. (b) Find an epression for Pr(K k), for k,,,, in erms of μ and k. Suppose ha μ.. (c) Find Pr(K ). (d) Find Pr(K ). Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

46 Chaper : Survival Disribuions C-3. Which of he following is equivalen o u pμ udu? (A) p (B) q (C) f () (D) f () (E) f ()μ 3. Which of he following is equivalen o d p d (A) p μ (B) μ (C) f () (D) μ (E) f ()μ 4. ( Nov #36) Given: (i) μ F e, (ii).4p.5 Calculae F. (A). (B).9 (C). (D).9 (E). 5. (CAS 4 Fall #7) Which of he following formulas could serve as a force of moraliy? (I) μ Bc, B >, C > (II) μ a(b ), a >, b > (III) μ ( ) 3, (A) (I) only (B) (II) only (C) (III) only (D) (I) and (II) only (E) (I) and (III) only? Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

47 C-4 Chaper : Survival Disribuions 6 ( Nov #) You are given he survival funcion S (), where (i) S (), e (ii) S (), 4.5 (iii) S (), 4.5 Calculae μ 4. (A).45 (B).55 (C).8 (D). (E). / 7. (CAS 4 Fall #8) Given () S, for, calculae he probabiliy ha a life age 36 will die beween ages 5 and 64. (A) Less han.5 (B) A leas.5, bu less han. (C) A leas., bu less han.5 (D) A leas.5, bu less han.3 (E) A leas.3 8. (7 May #) You are given: (i) 3p 7.95 (ii) p (iii) μ d. 7 7 Calculae 5 p 7. (A).85 (B).86 (C).87 (D).88 (E).89 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

48 Chaper : Survival Disribuions C-5 9. (5 May #33) You are given: Calculae 4 4 q 5. (A).38 (B).39 (C).4 (D).43 (E) < 6 μ.4 6 < 7. (4 Nov #4) For a populaion which conains equal numbers of males and females a birh: m (i) For males, μ., f (ii) For females, μ.8, Calculae q 6 for his populaion. (A).76 (B).8 (C).86 (D).9 (E).96. ( May #8) For a populaion of individuals, you are given: (i) Each individual has a consan force of moraliy. (ii) The forces of moraliy are uniformly disribued over he inerval (, ). Calculae he probabiliy ha an individual drawn a random from his populaion dies wihin one year. (A).37 (B).43 (C).5 (D).57 (E).63 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

49 C-6 Chaper : Survival Disribuions. [Srucural Quesion] The moraliy of a cerain populaion follows he De Moivre s Law; ha is μ, < ω. ω (a) Show ha he survival funcion for he age-a-deah random variable T is S ( ), < ω. ω (b) Verify ha he funcion in (a) is a valid survival funcion. (c) Show ha p, < ω, < ω. ω 3. [Srucural Quesion] The probabiliy densiy funcion for he fuure lifeime of a life age is given by α ( ) αλ f, α, λ > ( λ ) α (a) Show ha he survival funcion for a life age, S (), is (b) Derive an epression for μ. (c) Derive an epression for S (). (d) Using (b) and (c), or oherwise, find an epression for f (). α λ (. S ) λ 4. [Srucural Quesion] For each of he following equaions, deermine if i is correc or no. If i is correc, prove i. (a) u q p u q (b) u q q u q d (c) p p ( μ μ ) d Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

50 Chaper : Survival Disribuions C-7 Soluions o Eercise. (a) F (). d d ( ) ( ) (b) f() F() S( ) (c) () S. S ( ) (d) p S () /. (e) 5 q 3 p 3 5 p 3 S 3 () S 3 (5) S () 3 3 [(3 u) ] 3 (3 ) d 3 3 u u (3 ) f ( u)du If follows ha p 3 S(5 ) (3 5 ) 5 S5() 3 S (5) (3 5) 5. 3 [( u) ] ( )d u u ( ) 3. S () f ( u)du. 4 4 f ( ) μ. S ( ) ( ) 4 Hence, μ /( ).. 4. (a) Firs, noe ha μ. We have 8 S ( ) ep udu ep du μ 8 u 8 ep([ln(8 u)] ) ep(ln ). 8 8 (b) 4 p S (4) 4/8 /. (c) f () S ()μ Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

51 C-8 Chaper : Survival Disribuions 5. Our goal is o find Pr( < T < 5) S () S (5). Given he force of moraliy, we can find he survival funcion as follows: S ( ) ep d ep d μ u u u u ep([ln( u)] ) ep( ln ) So, he required probabiliy is ( /) ( 5/) α S ( ) ω ω μ α S ( ) ω α We are given ha μ 4 μ. This implies α. ω α α, which gives ω 6. ω 4 ω 7. (a) The probabiliy ha a new born infan dies no laer han age 35 can be epressed as 35 q. [Here we have q for a deah probabiliy, and 35.] Furher, 35 q F (35) S (35).65. (b) The probabiliy ha a person age now survives o age 5 can be epressed as 5 p. [Here we have p for a survival probabiliy, and 5 5.] S (5) Furher, we have 5 p S (5).977. S (5) (c) The probabiliy ha a person age 4 now survives o age 5 bu dies before aaining age 55 can be epressed as 5 q 4. [Here, we have q for a (deferred) deah probabiliy, 4, 5 4, and u ] S(5) Furher, we have 5 q 4 S 4 () S 4 (5) S (4) S (55).35. S (4) 8. The probabiliy ha a person aged will die beween he ages of 5 and 6 is given by 3 q 3 p 4 p S (3) S (4). S ( ) ( ) S. S () So, S (3), S (4). As a resul, 3 q 9/ Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

52 Chaper : Survival Disribuions C-9 9. (a) 3p p p (b) p p p q q p 3 3 p.9653 (c) 3q p 5 p.98 (.775) (a) Pr(K ) q p q ( q )q (.)..8 (b) Pr(K ) q. Pr(K ) q p q p p q ( q )( q )q Hence, Pr(K ) (a) Given ha μ μ for all, we have p e μ, p e μ and q e μ. (b) Pr(K k) k q k p e (k )μ. Pr(K k) k q k p q k e kμ ( e μ ). (c) When μ., Pr(K ) e. ( e. ).9. (d) When μ., Pr(K ) e ( )..4.. Firs of all, noe ha u p μ u in he inegral is simply f (u). u Hence, he answer is (B). pμ udu f ( u)du Pr( T ) F () q. 3. Mehod I: We use p q. Differeniaing boh sides wih respec o, d d p d d q F ( ) f ( ). d d Noing ha f () p μ, he answer is (A). Mehod II: We differeniae p wih respec o as follows: d d d ( ) ep d p S d d d μ u u d ep d d. μ u u d μ u u d Recall he fundamenal heorem of calculus, which says ha g( u)du g( ) d. Thus c d p ep u u p μ d ( μ ) μ. d Hence, he answer is (A). Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

53 C-3 Chaper : Survival Disribuions 4. Firs, noe ha The eponen in he above is u μud u ( F e )d u p e e..4 ( F e )du Fu e u u.4f f.677 As a resul,.5 e.4f.677, which gives F.. Hence, he answer is (E) Recall ha we require he force of moraliy o saisfy he following wo crieria: (i) μ for all, (ii) μ d. All hree specificaions of μ saisfy Crierion (i). We need o check Crierion (ii). We have and Bc Bc d, ln c a b d aln( b ), d. ( ) ( ) 3 Only he firs wo specificaions can saisfy Crierion (ii). Hence, he answer is (D). [Noe: μ Bc is acually he Gomperz law. If you knew ha you could have idenified ha μ Bc can serve as a force of moraliy wihou doing he inegraion.] S ( ) 6. Recall ha μ. S ( ) Since we need μ 4, we use he definiion of S () for 4.5: () e e S, S ( ). 4 e 4 As a resul, μ e e e. Hence, he answer is (E). Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

54 Chaper : Survival Disribuions C-3 7. The probabiliy ha a life age 36 will die beween ages 5 and 64 is given by S 36 (5) S 36 (8). We have S 36 / 36 / S (36 ) ( ) /. S (36) This gives S 36 (5) and S 36 (8). As a resul, he required probabiliy is 8 8 S 36 (5) S 36 (8) /8.5. Hence, he answer is (A). 8. The compuaion of 5 p 7 involves hree seps. 3p7.95 Firs, p p.96 Second, 7 75 d 7.7 4p 7 e μ e Finally, 5 p Hence, he answer is (E) p 5 e p 5 e p 6 e p 5 p 5 8 p Finally, 4 4 q 5 4 p 5 8 p Hence, he answer is (A).. For males, we have For females, we have For he overall populaion, and m S () e e e m μu du.du.. f S () e e e f μu du.8du e e S (6).5354, e e S (6).49. Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

55 C-3 Chaper : Survival Disribuions S(6) Finally, q6 p6.8. Hence, he answer is (B). S (6). Le M be he force of moraliy of an individual drawn a random, and T be he fuure lifeime of he individual. We are given ha M is uniformly disribued over (, ). So he densiy funcion for M is f M (μ) / for < μ < and oherwise. This gives Pr( T ) E[Pr( T M)] Pr( T M μ) f ( μ)dμ μ ( e ) dμ M Hence, he answer is (D). ( e ) ( e ) (a) We have, for < ω, ln( ) ω S ( ) ep( μ sds) ep( ds) ep([ln( ω s)] e ω s ω. (b) We need o check he following hree condiions: (c) (i) S () /ω (ii) S (ω) ω/ω (iii) S (ω) /ω < for all < ω, which implies S () is non-increasing. Hence, he funcion in (a) is a valid survival funcion. p S ( ) ω ω, for < ω, < ω. S ( ) ω ω ω α α S ( αλ λ ) F ( ) f ( s)ds ds. ) α f ( ) α (b) μ. S ( ) λ 3. (a) α ( λ s) ( λ Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

56 Ace Learning Johnny Li and Andrew Ng SoA Eam MLC C-33 Chaper : Survival Disribuions (c). ) ( ) ( ) ( α α α λ λ λ λ λ λ S S S (d) S f λ α λ λ μ α ) ( ) (. 4. (a) No, he equaion is no correc. The correc equaion should be u q p u q. (b) No, he equaion is no correc. The correc equaion should be u p p u p. (c) Yes, he equaion is correc. The proof is as follows: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )] ( [ )) ( ))( ( )) ( )( ( )] ( [ ) ( ' ) ( ) ( ' ) ( ) ( ) ( d d d d p p p S f S S S S S f S f S S S f S f S f S S S S S S S S p μ μ μ μ

57 C-34 Chaper : Survival Disribuions Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

58 Chaper : Life Tables C- Chaper Life Tables OBJECTIVES. To apply life ables. To undersand wo assumpions for fracional ages: uniform disribuion of deah and consan force of moraliy 3. To calculae momens for fuure lifeime random variables 4. To undersand and model he effec of selecion Acuaries use spreadshees eensively in pracice. I would be very helpful if we could epress survival disribuions in a abular form. Such ables, which are known as life ables, are he focus of his chaper.. Life Table Funcions Below is an ecerp of a (hypoheical) life able. In wha follows, we are going o define he funcions l and d, and eplain how hey are applied. l d Ace Learning Johnny Li and Andrew Ng SoA Eam MLC

59 C- Chaper : Life Tables In his hypoheical life able, he value of l is,. This saring value is called he radi of he life able. For,,., he funcion l sands for he epeced number of persons who can survive o age. Given an assumed value of l, we can epress any survival funcion S () in a abular form by using he relaion l l S (). In he oher way around, given he life able funcion l, we can easily obain values of S () for inegral values of using he relaion Furhermore, we have l S( ). l p S(), S( ) l / l l S ( ) l / l l which means ha we can calculae p for all inegral values of and from he life able funcion l. The difference l l is he epeced number of deahs over he age inerval of [, ). We denoe his by d. I immediaely follows ha d l l. We can hen calculae q and m n q by he following wo relaions: q d l l l, l l l d n m m m n m n q. l l When, we can omi he subscrip and wrie d as d. By definiion, we have l l Graphically, d d d... d. d d d d 3... d d l l age l l Ace Learning Johnny Li and Andrew Ng SoA Eam MLC