Summary of Formulae for Actuarial Life Contingencies

Transcription

1 Summary of Formulae for Actuarial Life Contingencies Contents Review of Basic Actuarial Functions... 3 Random Variables... 5 Future Lifetime (Continuous)... 5 Curtate Future Lifetime (Discrete) /m thly Curtate Future Lifetime (Discrete)... 5 Contingent Payment Models... 6 Whole life Insurance,,,,,... 6 Continuous Case... 6 Annual Case /m thly case Increasing Case Increasing continuous piece-wise Increasing continuous Term Insurance,,,,,,,, Continuous Case Annual Case /m thly case Increasing Case Increasing continuous piece-wise Increasing continuous Increasing Case Increasing continuous piece-wise Increasing continuous Pure endowment Both Continuous and Discrete case Endowment Insurance,, Review of Basic Actuarial Functions 1

2 Continuous Case Annual Case /m thly case Contingent Annuity Models Whole life Annuity Models,,,,,,,, Due Case Immediate Case Continuous Case /m thly Due Case /m thly Immediate Case Term/Temporary Annuity Models,,,,,,,, Due Case Due Case Continuous Case /m thly Due Case /m thly Immediate Case Summary Tables - Whole of Life Assurance Summary Tables - Term Assurance Summary Tables - Endowment Assurance Summary Tables - Whole of Life Annuity Summary Tables - Maximum-term Annuities Review of Basic Actuarial Functions 2

3 Review of Basic Actuarial Functions intervals with the first payment due at time at an effective interest rate of pa. intervals with the first payment due at time at an effective interest rate of pa. intervals with the first payment at time. intervals with the first payment at time. intervals with the first payment due at time. intervals with the first payment due at time. Is the present value of a continuous payment at the rate of 1 per time unit over time units with the payment period commencing at time 0. Is the value at time of a continuous payment at the rate of 1 per time unit over time units with the payment period commencing at time Is the present value of a continuous payment at the rate of 1 per time unit over time units with the payment period commencing at time Is the present value of a series of payments of at intervals of with the first payment due at time. Is the present value of a series of payments of at intervals of with the first payment due at time 0. Is the value at time of a series of payments of at intervals of with the first payment at time. Is the value at time of a series of payments of at intervals of with the first payment at time 0. Is the present value of a series of payments of at intervals of with the first payment due at time. Is the present value of a series of payments of at intervals of with the first payment due at time. Is the present value of a series of payments where the amount Is the present value of a series of payments where the amount Review of Basic Actuarial Functions 3

4 Is the present value of a series of payments where the amount Is the value at time of a series of payments where the amount Is the value at time of a series of payments where the amount Is the present value of a series of continuous payments where the rate of payment in the time interval is for Is the value at time of a series of continuous payments where the rate of payment in the time interval is for Is the present value of a continuous payment where the rate of payment at time in the interval is. Is the value at time of a continuous payment where the rate of payment at time in the time interval is Is the present value of a series of payments where the amount. Review of Basic Actuarial Functions 4

5 Random Variables Future Lifetime (Continuous) is the future lifetime of - a person aged exactly. Curtate Future Lifetime (Discrete) is the number of whole years lived by. and. indicates died in the interval for. for. 1/m thly Curtate Future Lifetime (Discrete) be the future lifetime of in years rounded to the lower 1/m th of a year., and. indicates died in the interval for for.. Random Variables 5

6 Contingent Payment Models Distribution Function Whole life Insurance,,,,, Continuous Case Description The present value of $1 paid immediately on death of. Random Variable (Continuous) Deferred The present value of $1 paid immediately on death of, provided death occurs after years. EPV (mixed) Distribution Function Moments (at n times the force of interest ). Variance Relationships Contingent Payment Models 6

7 Method 2. and are independent. UDD Method 1 (or let in.) Claims Acceleration Approach Under UDD of death over the year of age, average time of payment is 1/2 in the year of death. Contingent Payment Models 7

8 Deterministic Approach : premium each of the persons pay to purchase the insurance; each person's share of the total present value of all benefit payments. Contingent Payment Models 8

9 Annual Case Distribution Function Description The present value of $1 paid at the end of the year of death of. Random Variable (Discrete) EPV Deferred The present value of $1 paid at the end of the year of death of, provided death occurs after years. (discrete) Moments Relationships (at n times the force of interest ). Variance Evaluation 1. First Principle: Set 2. Recursion:, 3. Tables... Contingent Payment Models 9