Actuarial Methods in Life Insurance 2400-M1IiEMAŻ
1. Life contingencies (4 h.)
Basic financial mathematics: effective and nominal interest rates, financial annuities. Future lifetime as a key random variable. Survivorship functions. The force of mortality. Analytical distributions of death. Probability of death for fractions of a year. Empirical mortality tables. (Skałba chap. 2; Bowers chap. 3; Wiśniewski chap. 1)
2. Net present value of a life and endowment contract (4 h.)
Amount payable at death as a random variable. Elementary life and endowment contracts. Continuous vs. discrete model. Variable life policies. Recursive formulae. (Skałba chap. 3; Gerber chap. 3; Bowers chap. 4; Wiśniewski chap. 2)
3. Net present value of a life annuity (4 h.)
Main types of life annuities. Continuous vs. discrete model. Annuities with payments made m times in a year. Variable annuities. Actuarial equivalence equations for life and annuities contracts. Commutations functions. (Skałba chap. 4; Gerber chap. 4; Bowers chap. 5; Wiśniewski chap. 3)
4. Annuitization of net single premiums (4 h.)
Policy loss function and actuarial equivalence principle. Continuous vs. discrete model of premium payments. Premiums payed m times a year. A general type of life insurance. (Skałba chap. 5; Gerber chap. 5; Bowers chap. 6; Wiśniewski chap. 4)
5. Net premium reserves (4 h.)
Prospective vs. retrospective concept of reserves. Recursive formulae of reserves. Sum at risk and survival risk. Universal life contracts and conversion of an insurance. Technical gain of a policy year and its allocation. Policies linked to investment fund units. Commutation formulae for premiums and reserves. (Skałba chap. 6; Gerber chap. 6; Bowers chap. 7; Wiśniewski chap. 5)
6. Gross premiums and reserves (3 h.)
Basic types of expense loading. The expense-loaded premiums. Gross premium reserves. Zillmer’s approach to acquisition costs. Gross reserve vs insurer’s gain. Allocation of investment gains to policy. Policy’s cash value. Actuarial equivalence by policy alternations and conversions. (Skałba chap. 9; Gerber chap. 10; Wiśniewski chap. 6)
7. Extra mortality risks (2 h.)
A constant addition to the force of mortality. A proportional addition to the force of mortality. Risks operating at specific point of time. (Wiśniewski chap. 7)
8. Multiple-life insurance (4 h.)
Multiple-life statuses. The joint-life status. The last-survivor status. Premium formulae for the joint-life and the last-survivor status. The Schuette-Nesbitt formula. The general symmetric status of k survivors. Asymmetric statuses. Relevant examples of multiple life contracts: reversions and orphan’s annuities. (Skałba chap. 8; Gerber chap. 8; Bowers chap. 9)
9. Multiple-decrement model (3 h.)
The remaining lifetime of the current status under the multiple-decrement model of mutually exclusive decriments. Forces of decriments. Continouos vs. discrete model. Multiple-decriments tables. Decrements for fractions of a year. Premiums and reserves in the multiple-decriments insurance. (Skałba chap. 7; Gerber chap.7; Bowers chap. 9)
Type of course
Course coordinators
Learning outcomes
Knowledge:
The student is acquainted with modelling of cashflows resulting from a long-term contract with random duration. He/she knows methods of modelling the demography, which describes the distribution of a future lifetime of the contracts. She/he identifies the situations where either continuous- or discrete-time modelling is appropriate and knows how to bridge the both techniques. He/she understands a need to recognize the time-value of money and to apply a present value approach to modelling. The student knows several types of long-term contracts in which an increasing monotonic hazard function operates, thus reflecting the demography of human beings. She/he knows the basic concepts of an actuarial equivalence, i.e. concepts of balancing the gains and losses for the both parties of the contract. He/she knows the methods to determine the current value of the contract. The student is acquainted with equivalent methods of alternations and conversions of the original terms of a contract. He/she knows the rules to maintain the contract’s solvency by adequate long-term reserves, balancing expected expenses with expected revenues. He is trained to extend the basic demography of a binary status (life or death) to the case of a small group of individuals/objects with binary status or to the case of single individuals/objects with multiple exits from the current state.
Abilities:
The student is able to apply a bundle of basic actuarial methods dedicated to life insurance. He/she is able to examine the balance of cashflows in a long-term contract with random duration. She/he can formulate the equivalence equation for the starting moment of the contract, as well as for whatever time in the future. He/she is prepared to reckon the expected gain/loss from the contract and its variance.
The student is able to evaluate possible alternations and conversions introduced to the contract after its inception. She/he can calculate the current cash value of the contract and evaluate alternative methods to allocate the investment bonus to a given insurance policy.
Social competences
While understanding the complex nature of a life insurance contract, the student may contribute to the public discussion wherever a long-term sharing of risk is on debate. Thus he/she may strengthen the rational attitude toward the markets where risks are transferred in a long-run perspective.
KW01, KW02, KW03, KW04, KU01, KU02, KU03, KU04, KU05, KU06, KU07, KK01, KK02, KK03
Assessment criteria
Students are supposed to submit two class exercise forms and to write the final examination. The total score constitutes of class exercises (max. 20 points each) and exam exercises (max. 40 points). Required minimum amounts to 41 points in total.
Bibliography
Textbooks
1. Bowers Newton L., Hans U. Gerber, James C. Hickman, Donald A. Jones, Cecil J. Nesbitt, Actuarial Mathematics, Society of Actuaries, Itasca 1986.
2. Gerber Hans U., Life Insurance Mathematics, Springer, Berlin 1995.
3. Skałba Mariusz, Ubezpieczenia na życie, WNT Warszawa 1999.
4. Marian Wiśniewski, Metody aktuarialne w ubezpieczeniach na życie, WNE UW 2018, skrypt.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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