ACTEX Exam MLC Webcast/DVD Seminar Handout

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Transcript of ACTEX Exam MLC Webcast/DVD Seminar Handout

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

    FROM: Richard L. (Dick) London, FSA

    Dear Students,

    Thank you for purchasing the DVD recording of the ACTEX Review Seminar for SOA Exam M,Life Contingencies segment (MLC). This version is intended for the exam offered in May 2007and thereafter. Please be aware that the DVD seminar does not deal with the FinancialEconomics segment (MFE) of Exam M.

    The purpose of this memo is to provide you with an orientation to this seminar, which isdevoted to a review of life contingencies plus the related topics of multi-state models and thePoisson process. A 6-page summary of the topics covered in the seminar is attached to this covermemo. Although the seminar is organized independently of any particular textbook, thesummary of topics shows where each topic is covered in the textbook Models for Quantifying Risk(Second Edition), by Cunningham, Herzog, and London.

    In order to be ready to write the MLC exam, you need to accomplish three stages of preparation:

    This first stage is to obtain an understanding of all the underlying mathematical theory, includinga mastery of standardized international actuarial notation. This DVD review seminar is designedto enable you to obtain that understanding.

    The second stage is to prepare, and then master, a complete list of all formulas and relationships(of which there are many) needed for the exam. For those who do not wish to prepare their ownlists, a very complete set of formulas (364 in total) is available from ACTEX in the form of studyflash cards.

    The third stage is to do a large number of exam-type practice questions, beginning with the end-of-chapter exercises is the textbook. In addition, you should purchase one (or more) of theseveral exam-prep study guides that have been prepared for that purpose. A relatively smallnumber of sample problems (34) are worked out for the group in this DVD. A copy of thesequestions, with detailed solutions, is attached to this cover memo for your reference.

    Good luck to you on your exam

  • -1-

    Summary of Topics

    A. Parametric Survival Models

    1. Age-at-Failure Random Variable X (3.1)a. CDFb. SDFc. PDFd. HRFe. CHFf. Momentsg. Actuarial notation and terminology

    2. Examples (3.2)a. Uniformb. Exponentialc. Gompertzd. Makehame. Weibullf. Others via transformation

    3. Time-to-Failure Random Variable xT (3.3)

    a. SDFb. CDFc. PDFd. HRFe. Momentsf. Discrete counterpart xK

    g. Curtate duration K(x)

    4. Central Rate (3.4)

    5. Select Models (3.5)

    B. The Life Table

    1. Definition (4.1)

    2. Traditional Form (4.2)a. x

    b. xd and n xd

    c. xp and n xp

    d. xq and n xq

    3. Other Functions (4.3)a. x and x t b. PDF and moments of X

  • -2-

    c. |n m xq

    d. PDF and moments of xT

    e. Temporary expectationf. Curtate expectationg. Temporary curtate expectationh. Central rate

    4. Non-Integral Ages (4.5)a. Linear assumptionb. Exponential assumptionc. Hyperbolic assumption

    5. Select Tables (4.6)

    C. Contingent Payment Models

    1. Discrete Models (5.1)a. ,xZ and its moments

    b. 1:x nZ and its moments

    c. | ,n xZ and its moments; covariance with1:x nZ


    :x nZ and its moments; covariance with1:x nZ

    e. : ,x nZ and its moments

    2. Group Deterministic Interpretation (5.2)

    3. Continuous Models (5.3)

    a. ,xZ and its moments

    b. 1:x nZ and | ,n xZ and their moments

    c. :x nZ

    d. ( ) ,mxZ and its moments

    e. Evaluation under exponential distributionf. Evaluation under uniform distribution

    4. Varying Payments (5.4)

    a. xB in general

    b. (IA)x

    c. 1:( )x nIA and1

    :( )x nDA

    d. ( ) ,xIA1

    :( )x nIA and1

    :( )x nDA

    e. ( ) ,xI A1

    :( ) ,x nI A and1

    :( ) ,x nDA

    5. Approximation from Life Table (5.5)a. Continuous modelsb. mthly models

  • -3-

    D. Contingent Annuity Models

    1. Whole Life (6.1)a. Immediateb. Duec. Continuous

    2. Temporary (6.2)a. Immediateb. Duec. Continuous

    3. Deferred (6.3)a. Immediateb. Duec. Continuous

    4. mthly Payments (6.4)a. Immediateb. Duec. Random variablesd. Approximation from life table

    5. Non-Level Payments (6.5)a. Immediateb. Duec. Continuous

    E. Funding Plans

    1. Annual Payment Funding (7.1)a. Discrete payment modelsb. Continuous payment modelsc. Non-level funding

    2. Random Variable Analysis (7.2)d. Present value of loss random variablee. Expected valuef. Variance

    3. Continuous Payment Funding (7.3)a. Discrete payment modelsb. Continuous payment models

    4. mthly Payment Funding (7.4)a. Discrete payment modelsb. Continuous payment models

    5. Incorporation of Expenses (7.5)

  • -4-

    F. Reserves

    1. Annual Payment Funding (8.1)a. Prospective methodb. Retrospective methodc. Additional expressionsd. Random variable analysise. Continuous payment modelsf. Contingent annuity model

    2. Recursive Relationships (8.2)a. Group deterministic analysisb. Random variable analysis cash basisc. Random variable analysis accrued basis

    3. Continuous Payment Funding (8.3)a. Discrete payment modelsb. Continuous payment modelsc. Random variable analysis

    4. mthly Payment Funding (8.4)

    5. Incorporation of Expenses (8.5)

    6. Fractional Duration Reserves (8.6)

    7. Non-Level Benefits and Premiums (8.7)a. Discrete modelsb. Continuous models

    G. Multi-Life Models

    1. Joint-Life Model (9.1)a. Random variable xyT

    b. SDFc. CDFd. PDFe. HRFf. Conditional probabilitiesg. Moments

    2. Last-Survivor Model (9.2)

    a. Random variable xyT

    b. CDFc. SDFd. PDFe. HRFf. Moments

    g. Relationships of xyT and xyT

  • -5-

    3. Contingent Probability Functions (9.3)

    4. Contingent Multi-Life Contracts (9.4)a. Contingent payment modelsb. Contingent annuity modelsc. Premiums and reservesd. Reversionary annuitiese. Contingent insurance functions

    5. Random Variable Analysis (9.5)a. Marginal distributionsb. Covariancec. Joint functionsd. Joint-lifee. Last-survivor

    6. Common Shock (9.6)

    H. Multiple-Decrement Models

    1. Discrete Models (10.1)a. Multiple-decrement tablesb. Random variable analysis

    2. Competing Risks (10.2)

    3. Continuous Models (10.3)

    4. Uniform Distribution (10.4)a. Multiple-decrement contextb. Single-decrement context

    5. Actuarial Present Value (10.5)

    6. Asset Shares (10.6)

    I. Poisson Process

    1. Properties (11.3.1-11.3.3)

    2. Mixture Process (11.3.4)

    3. Nonstationary Process (11.3.5)

    4. Compound Poisson Process (14.1)

  • -6-

    J. Multi-State Models

    1. Review of Markov Chains (Appendix A)

    2. Homogeneous Multi-State Process (10.7.1)

    3. Nonhomogeneous Multi-State Process (10.7.2)

    4. Daniels Notation (SN: M-24-05)

  • -7-

    Practice Questions

    1. Let a survival distribution be defined by 2( ) ,XS x ax b for 0 .x k If the expected value of X

    is 60, find the median of X.

    2. Given that ,x kx for all 0,x and 10 35 .81,p find the value of 20 40.p

    3. If X has a uniform distribution over (0, ), show that1 .50




    for 1.x

    4. Given that 0e 25 and ,x x for 0 ,x find the value of 10( ),Var T where 10T

    denotes the future lifetime random variable for an entity known to exist at age 10.

    5. Given the UDD assumption and the values 80.5 .0202, 81.5 .0408, and 82.5 .0619, find

    the value of 2 80.5.q

    6. Let x k denote a constant force of mortality for the age interval ( , 1).x k x k Find the value

    of :3 ,xe the expected number of years to be lived over the next three years by a life age x, given

    the following data:

    k x ke

    1 x k

    x k


    0 .9512 .97541 .9493 .97442 .9465 .9730

    7. Given the following excerpt from a select and ultimate table with a two-year select period, and

    assuming UDD between integral ages, find the value of .90 [60] .60 .q

    x [ ]x [ ] 1x 2x 2x

    60 80,625 79,954 78,839 6261 79,137 78,402 77,252 6362 77,575 76,770 75,578 64

    8. Calculate the value of 51( ),Var Z given the following values:

    51 50

    2 251 50 50

    .004 .02

    .005 .98

    A A i

    A A p

  • -8-

    9. Let the age-at-failure random variable X have a uniform distribution with 110. Let ( )Zf z denote

    the PDF of the random variable 40.Z Calculate the value of (.80),Zf given also that .05.

    10. (a) Show that 1 1( ) ( ) .x x x xIA A E IA

    (b) Calculate the value of 36( ) ,IA given the following values:

    35 35:1

    35 35

    ( ) 3.711 .9434

    .1300 .9964

    IA A

    A p

    11. Assuming failures are uniformly distributed over each interval ( , 1),x x calculate the value of2 1

    : 2xA given the values .12,i .10,xq and 1 .20.xq

    12. Find the value of ,xA given the values 10,xa 2 7.375,xa and ( ) 50.

    xTVar a

    13. Let S denote the number of annuity payments actually made under a unit 5-year deferred wholelife annuity-due. Find the value of 5( | ),xPr S a given the following values:


    xa .04i .01,x t for all t

    14. Show that, under the UDD assumption( )

    .xxi i d a


    15. Calculate the probability that the present value of payments actually made under a unit 3-yeartemporary increasing annuity-due will exceed the APV of the annuity contract, given thefollowing values:

    .80xp 1 .75xp 2 .50xp .90v

    16. Calculate the value of :1000 ( ),x nP A assuming UDD over each interval ( , 1)x x and the

    following values:

    : .804 .600 .04n xx nA E i

    17. For a unit whole life insurance, let xL denote the present value of loss r