Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014
Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value cashflows depedet o death, survival, or other ucertai risks. Liks to other subjects Subjects CT1 Fiacial Mathematics, CT3 Probability ad Mathematical Statistics ad CT4 Models: itroduce techiques that will be draw upo ad used i the developmet of this subject. Subjects ST1 Health ad Care Specialist Techical, ST2 Life Isurace Specialist Techical ad ST4 Pesios ad other Beefits Specialist Techical: use the priciples itroduced i this subject. Objectives O completio of this subject the cadidate will be able to: (i) Defie simple assurace ad auity cotracts, ad develop formulae for the meas ad variaces of the preset values of the paymets uder these cotracts, assumig costat determiistic iterest. 1. Defie the followig terms: whole life assurace term assurace pure edowmet edowmet assurace whole life level auity temporary level auity guarateed level auity premium beefit icludig assurace ad auity cotracts where the beefits are deferred. 2. Defie the followig probabilities: m q x, q x ad their select equivalets m q [x]+r, q [x]+r. 3. Obtai expressios i the form of sums for the mea ad variace of the preset value of beefit paymets uder each cotract above, i terms of the curtate radom future lifetime, assumig that death beefits are payable at the ed of the year of death ad that auities are paid aually i advace or i arrear, ad, where appropriate, simplify these expressios ito a form suitable for evaluatio by table look-up or other meas. 4. Obtai expressios i the form of itegrals for the mea ad variace of the preset value of beefit paymets uder each cotract above, i terms of the radom future Page 2 Istitute ad Faculty of Actuaries
lifetime, assumig that death beefits are payable at the momet of death ad that auities are paid cotiuously, ad, where appropriate, simplify these expressios ito a form suitable for evaluatio by table look-up or other meas. 5. Defie the symbols A x, A, 1 1 A, A, a x, a, : m a x, a x, a, m a ad their select ad cotiuous equivalets. Exted the auity factors to allow for the possibility that paymets are more frequet tha aual but less frequet tha cotiuous. 6. Uderstad ad use the relatios betwee auities payable i advace ad i arrear, ad betwee temporary, deferred ad whole life auities. 7. Uderstad ad use the relatios A x = 1 da x, cotiuous equivalets. A = 1 da, ad their select ad 8. Defie the expected accumulatio of the beefits i 1., ad obtai expressios for them correspodig to the expected preset values i 3. ad 4. (ote: expected values oly). (ii) Describe ad use practical methods of evaluatig expected values ad variaces of the simple cotracts defied i objective (i). 1. Describe the life table fuctios l x ad d x ad their select equivalets l [x]+r ad d [x]+r. 2. Express the followig life table probabilities i terms of the fuctios i 1.: p x, q x, m q x ad their select equivalets p [x]+r, q [x]+r, m q [x]+r. 3. Express the expected values ad variaces i objective (i) 3. i terms of the fuctios i 1. ad 2. 4. Evaluate the expected values ad variaces i objective (i) 3. by table look-up or other meas, icludig the use of the relatioships i objectives (i) 6. ad 7. 5. Derive approximatios for, ad hece evaluate, the expected values ad variaces i objective (i) 4. i terms of those i objective (i) 3. 6. Evaluate the expected accumulatios i objective (i) 8. (iii) Describe ad calculate, usig ultimate or select mortality, et premiums ad et premium reserves of simple isurace cotracts. 1. Defie the et radom future loss uder a isurace cotract, ad state the priciple of equivalece. 2. Defie ad calculate et premiums for the isurace cotract beefits i objective (i) 1. Regular premiums ad auity beefits may be payable aually, more frequetly tha Istitute ad Faculty of Actuaries Page 3
aually, or cotiuously. Death beefits may be payable at the ed of the year of death, or immediately o death. 3. State why a isurace compay will set up reserves. 4. Describe prospective ad retrospective reserves. 5. Defie ad evaluate prospective ad retrospective et premium reserves i respect of the cotracts i objective (i) 1., with premiums as i (iii) 2. 6. Show that prospective ad retrospective reserves are equal whe calculated o the same basis. 7. Obtai recursive relatioships betwee et premium reserves at aual itervals, for cotracts with death beefits paid at the ed of the year of death, ad aual premiums. 8. Defie ad calculate, for a sigle policy or a portfolio of policies (as appropriate): death strai at risk expected death strai actual death strai mortality profit for policies with death beefits payable immediately o death or at the ed of the year of death; for policies payig auity beefits at the start of the year or o survival to the ed of the year; ad for policies where sigle or aual premiums are payable. (iv) Describe ad calculate, usig ultimate or select mortality, et premiums ad et premium reserves for icreasig ad decreasig beefits ad auities. 1. Exted the techiques of (ii) to calculate the expected preset value of a auity, premium, or beefit payable o death, which icreases or decreases by a costat compoud rate. Calculate et premiums ad et premium reserves for cotracts with premiums ad beefits which vary as described. 2. Defie the symbols (IA) x, (IĀ) x, ( Ia ) x, (Ia) x ad (Iā) x ad their select equivalets. 3. Calculate the expected preset value of a auity, premium or beefit payable o death, which icreases or decreases by a costat moetary amout. Calculate et premiums ad et premium reserves for cotracts with premiums ad beefits which vary as described. 4. Describe the operatio of covetioal with-profits cotracts, i which profits are distributed by the use of regular reversioary bouses, ad by termial bouses. 5. Calculate et premiums ad et premium reserves for the covetioal with profits cotracts defied i 4. Page 4 Istitute ad Faculty of Actuaries
6. Describe the operatio of accumulatig with-profits cotracts, i which beefits take the form of a accumulatig fud of premiums, where either: the fud is defied i moetary terms, has o explicit charges, ad is icreased by the additio of regular guarateed ad bous iterest paymets plus a termial bous; or the fud is defied i terms of the value of a uit fud, is subject to explicit charges, ad is icreased by regular bous additios plus a termial bous ( Uitised with-profits ) I the case of uitised with-profits, the regular additios ca take the form of (a) uit price icreases (guarateed ad/or discretioary), or (b) allocatios of additioal uits. I either case a guarateed miimum moetary death beefit may be applied. (v) Describe ad calculate gross premiums ad reserves of assurace ad auity cotracts. 1. List the types of expeses icurred i writig a life isurace cotract. 2. Describe the ifluece of iflatio o the expeses listed i 1. 3. Defie the gross future loss radom variable for the beefits ad auities listed i (i) 1. ad (iv) 1. 5., ad calculate gross premiums ad reserves that satisfy probabilities ivolvig the future loss radom variable. Regular premiums ad auity beefits may be payable aually or cotiuously. Death beefits may be payable at the ed of the year of death or immediately o death. 4. Calculate gross premiums usig the equivalece priciple. Regular premiums ad auity beefits may be payable aually, more frequetly tha aually, or cotiuously. Death beefits may be payable at the ed of the year of death or immediately o death. 5. Calculate gross premiums usig simple criteria other tha the priciples described i 3. ad 4. 6. Defie ad calculate the gross premium prospective reserve. 7. Defie ad calculate the gross premium retrospective reserve. 8. State the coditios uder which, i geeral, the prospective reserve is equal to the retrospective reserve allowig for expeses. 9. Prove that, uder the appropriate coditios, the prospective reserve is equal to the retrospective reserve, with or without allowace for expeses, for all stadard fixed beefit ad icreasig/decreasig beefit cotracts. Istitute ad Faculty of Actuaries Page 5
10. Obtai a recursive relatio betwee successive aual reserves for a aual premium cotract, with allowace for expeses, for stadard fixed beefit cotracts, ad use this relatio to calculate the profit eared from a cotract durig a year. (vi) Defie ad use fuctios ivolvig two lives. 1. Exted the techiques of objectives (i) (v) to deal with cashflows depedet upo the death or survival of either or both of two lives. 2. Exted the techiques of 1. to deal with fuctios depedet upo a fixed term as well as age. (vii) Describe ad illustrate methods of valuig cashflows that are cotiget upo multiple trasitio evets. 1. Defie health isurace, ad describe simple health isurace premium ad beefit structures. 2. Explai how a cashflow, cotiget upo multiple trasitio evets, may be valued usig a multiple-state Markov Model, i terms of the forces ad probabilities of trasitio. 3. Costruct formulae for the expected preset values of cashflows that are cotiget upo multiple trasitio evets, icludig simple health isurace premiums ad beefits, ad calculate these i simple cases. Regular premiums ad sickess beefits are payable cotiuously ad assurace beefits are payable immediately o trasitio. (viii) Describe ad use methods of projectig ad valuig expected cashflows that are cotiget upo multiple decremet evets. 1. Defie a multiple decremet model as a special case of multiple-state Markov model. 2. Derive depedet probabilities for a multiple decremet model i terms of give forces of trasitio, assumig forces of trasitio are costat over sigle years of age. 3. Derive forces of trasitio from give depedet probabilities, assumig forces of trasitio are costat over sigle years of age. 4. Describe the costructio ad use of multiple decremet tables. 5. Describe the typical beefit ad cotributio structures of pesio schemes, icludig: defied cotributio schemes defied beefit (fial salary) schemes 6. Use multiple decremet tables to evaluate expected preset values of cashflows depedet upo more tha oe decremet, icludig those of pesio schemes. Page 6 Istitute ad Faculty of Actuaries
(ix) Describe ad use projected cashflow techiques, where ad as appropriate for use i pricig, reservig, ad assessig profitability. 1. Defie uit-liked cotract. 2. Project expected future cashflows for whole life, edowmet ad term assuraces, auities, uit-liked cotracts, ad uitised with-profits cotracts, icorporatig multiple decremet models as appropriate. 3. Profit test simple aual premium cotracts of the types listed i 2. ad determie the profit vector, the profit sigature, the et preset value, ad the profit margi. 4. Show how a profit test may be used to price a product, ad use a profit test to calculate a premium for a covetioal (without profits) life isurace cotract. 5. Show how, for uit-liked cotracts, o-uit reserves ca be established to elimiate ( zeroise ) future egative cashflows, usig a profit test model. (x) Describe the pricipal forms of heterogeeity withi a populatio ad the ways i which selectio ca occur. 1. Explai why it is ecessary to have differet mortality tables for differet classes of lives. 2. Explai the theoretical basis of the use of risk classificatio i life isurace. 3. State the pricipal factors which cotribute to the variatio i mortality ad morbidity by regio ad accordig to the social ad ecoomic eviromet, specifically: occupatio utritio housig climate/geography educatio geetics 4. Defie ad give examples of the mai forms of selectio: temporary iitial selectio class selectio time selectio spurious selectio adverse selectio 5. Explai how selectio ca be expected to occur amogst idividuals or groups takig out each of the mai types of life isurace cotracts, or amogst members of large pesio schemes. Istitute ad Faculty of Actuaries Page 7
6. Explai the cocept of mortality covergece 7. Explai how decremets ca have a selective effect. 8. Explai the cocept of a sigle figure idex ad its advatages ad disadvatages for summarisig ad comparig actual experiece. 9. Defie the terms crude mortality rate, directly stadardised ad idirectly stadardised mortality rate, stadardised mortality ratio, ad illustrate their use. END OF SYLLABUS Page 8 Istitute ad Faculty of Actuaries