Subject CT5 Contingencies Core Technical. Syllabus. for the 2011 Examinations. The Faculty of Actuaries and Institute of Actuaries.

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1 Subject CT5 Cotigecies Core Techical Syllabus for the 2011 Examiatios 1 Jue 2010 The Faculty of Actuaries ad Istitute of Actuaries

2 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. Subject ST2 Life Isurace Specialist Techical: uses the priciples ad techiques i this subject to help i the solutio of life isurace problems. 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 critical illess assurace whole life level auity temporary 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. Page 2

3 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 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. Exted the techiques of 3. ad 4. above to deal with the possibility that premiums are payable more frequetly tha aually ad that beefits may be payable aually or more frequetly tha aually. 6. 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. 7. Derive relatios betwee auities payable i advace ad i arrear, ad betwee temporary, deferred ad whole life auities. 8. Derive the relatios A x = 1 da x, equivalets. A = 1 da, ad their select ad cotiuous 9. Defie the expected accumulatio of the beefits i 1., ad obtai expressios for them correspodig to the expected preset values i 3., 4., ad 5. (ote: expected values oly). (ii) Describe 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. Page 3

4 4. Evaluate the expected values ad variaces i objective (i) 3. by table look-up, where appropriate, icludig the use of the relatioships i objectives (i) 7. ad Derive approximatios for, ad hece evaluate, the expected values ad variaces i objective (i) 4. i terms of those i objective (i) Evaluate the expected accumulatios i objective (i) Describe practical alteratives to the life table which ca be used to obtai the evaluatios i 4., 5., ad 6. (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. Premiums ad auities may be payable aually, more frequetly tha aually, or cotiuously. Beefits may be payable at the ed of the year of death, immediately o death, aually, more frequetly tha aually, or cotiuously. 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) Show that prospective ad retrospective reserves are equal whe calculated o the same basis. 7. Derive 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. Derive Thiele s differetial equatio, satisfied by et premium reserves for cotracts with death beefits paid at the momet of death, ad premiums payable cotiuously. 9. 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 Page 4

5 (iv) Describe the calculatio, usig ultimate or select mortality, of 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, ( Ia ) x, ad (Ia) 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. Defie with profits cotract. 5. List the types of bous that may be give to with profits cotracts. 6. Calculate et premiums ad et premium reserves for with profits cotracts. (v) Describe the calculatio of 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 Defie the gross future loss radom variable for the beefits ad auities listed i (i) 1. ad (iv). 4. Calculate gross premiums usig the future loss radom variable ad the equivalece priciple. Premiums ad auities may be payable aually, more frequetly tha aually, or cotiuously. Beefits may be payable at the ed of the year of death, immediately o death, aually, more frequetly tha aually, or cotiuously. 5. Calculate gross premiums usig simple criteria other tha the equivalece priciple. 6. Calculate gross premium prospective reserves usig the future loss radom variable. 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. Page 5

6 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. 10. Derive a recursive relatio betwee successive aual reserves for a aual premium cotract, with allowace for expeses, for stadard fixed beefit cotracts. (vi) Defie ad use straightforward 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 methods which ca be used to model cashflows cotiget upo competig risks. 1. Explai how the value of a cashflow, cotiget upo more tha oe risk, may be valued usig a multiple-state Markov Model. 2. Derive depedet probabilities from give trasitio itesities, usig the Kolmogorov equatios itroduced i Objective (vii) of the Models Subject. 3. Derive trasitio itesities from give depedet probabilities. (viii) Describe the techique of discouted emergig costs, for use i pricig, reservig, ad assessig profitability. 1. Defie uit-liked cotract. 2. Evaluate expected cashflows for whole life, edowmet ad term assuraces, auities, ad uit-liked cotracts. 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 the profit test may be used to price a product. 5. Show how the profit test may be used to determie reserves. 6. Describe the costructio ad use of multiple decremet tables, icludig the relatioships with associated sigle decremet tables. Page 6

7 7. Use multiple decremet tables to evaluate expected cashflows depedet upo more tha oe decremet, icludig: pesio beefits other salary related beefits health ad care isurace 8. Describe practical alteratives to the multiple decremet table which ca be used to obtai the evaluatios i Exted the techiques of 3., 6., ad 7. to evaluate expected cashflows cotiget upo risks other tha huma life. (ix) Describe the pricipal forms of heterogeeity withi a populatio ad the ways i which selectio ca occur. 1. 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 2. Defie ad give examples of the mai forms of selectio: temporary iitial selectio class selectio time selectio spurious selectio adverse selectio 3. Explai how selectio ca be expected to occur amogst idividuals takig out each of the mai types of life isurace cotracts, or amogst members of large pesio schemes. 4. Explai why it is ecessary to have differet mortality tables for differet classes of lives. 5. Explai how decremets ca have a selective effect. 6. Explai the theoretical basis of the use of risk classificatio i life isurace. Page 7

8 7. Explai the impact of the availability of geetic iformatio o risk classificatio i life isurace. 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. Ed of Syllabus Page 8

Institute of Actuaries of India Subject CT5 General Insurance, Life and Health Contingencies

Institute of Actuaries of India Subject CT5 General Insurance, Life and Health Contingencies Istitute of Actuaries of Idia Subject CT5 Geeral Isurace, Life ad Health Cotigecies For 2017 Examiatios Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which