## LIFE INSURANCE

## Program

EN
IT

### Updated A.Y. 2022-2023

**Program**

The provisional Program of the course is:

1.Reminder on Compound Interest

2.Survival probability: deterministic and stochastic models

3.Life Insurance contracts

4.Life Annuities

5.Net Premiums and Reserve Calculations

6.Unit linked life insurance contracts

The final program will be available at the end of the course.

**Reference Books:**

Life insurance mathematics. Hans U. Gerber. 3rd ed. Springer, 1997.

Actuarial Mathematics for Life Contingent Risks. David C. M. Dickson, Mary R. Hardy, Howard R. Waters. 3rd ed. Cambridge University Press, 2020.

**Detailed Syllabus**

- Reminder on Mathematical Finance:
- Compound interest rate, discount factor, present value
- Nominal interest rate and its properties
- Continuously compounded interest rates, force of interest
- Payments made with continuous rate of payments
- Interests credited in advance
- Perpetuities: due, immediate, present value, one payment per year, more than one payment per year, constant payments, variable payments
- Annuities: due, immediate, present value, accumulated value, one payment per year, more than one payment per year, constant payments, variable payments

- Reminder on probability:
- Random variables (discrete and continuous), Cumulative distribution function and its properties, expected value, variance and their properties.
- Conditional probability, Bayes rule, full probability rule, Conditional distribution

- Remaining lifetime of a policy holder, lifetime distribution, survival probability and their properties.
- Force of mortality and famous mortality laws (De Moivre, Constant force of mortality, Gompertz, Makeham, Weibull)
- Mean and Standard deviation of the remaining lifetime, Curtate lifetime, Fractional lifetime, usual assumptions on the fractional lifetime.
- Life tables
- Life insurance market and life insurance products: main characteristics.
- Actuarial value of standard life insurance products (life insurance and life annuities).
- Whole life insurance, term insurance, pure endowment, endowment insurance: actuarial values and properties
- Whole life annuity, term annuity, deferred annuities: actuarial value and properties. Annuities paid more than once per year. Whoolhouses’ formula.

- The cost of additional options and guarantees.
- Loss at issue on a policy (net/gross). Premiums (net and gross). The equivalence principle and the portfolio percentile premium principle for premiums calculation.
- Loss at time t, Policy value and its interpretation
- Net Amount at Risk and the premium decomposition: the saving premium and the risk premium
- Pricing of unit linked life insurance contracts