Finance and insurance

MARIA GIUSEPPINA BRUNO

The course aims to provide students with a solid understanding of the main methods and models of actuarial mathematics and their applications in insurance, with particular focus on life and non-life contracts. Students will acquire the knowledge necessary to calculate premiums and reserves for traditional insurance contracts, understand the strategic variables underlying coverage and portfolio diversification, and apply probabilistic and statistical techniques for risk measurement. The course integrates theoretical treatment with practical examples, potentially supported by computational tools, enabling students to develop operational skills in analyzing insurance contracts, evaluating risk measures, and designing products with flexible benefits or alternative coverage. Students will learn to critically interpret actuarial models, select the most appropriate methodologies for specific insurance problems, and clearly and rigorously present results, both orally and in writing. By the end of the course, students will have developed independent judgment in risk analysis and management, the ability to apply actuarial models critically to real scenarios, communication skills appropriate to the professional context, and autonomous learning methods useful for continuous deepening of advanced actuarial mathematics topics and for subsequent courses such as Risk Theory, Actuarial Theory and Techniques for Pension, and Insurance Techniques and Finance.

The course aims to provide students with a solid understanding of the main methods and models of actuarial mathematics and their applications in insurance, with particular focus on life and non-life contracts.

Knowledge and understanding: Students will acquire a solid understanding of the fundamental principles of actuarial mathematics and their applications in insurance, including the main types of life and non-life policies. They will be able to describe concepts such as random financial operation, expected present value, fairness, and expected utility; apply probabilistic methods for calculating premiums, reserves, and claims; understand mutuality and diversification of insurance portfolios; and evaluate operational strategies related to structured policies, financial options, and alternative reinsurance coverage.

Applying knowledge and understanding: Students will be able to calculate premiums and reserves for main life and non-life insurance contracts, discussing the results in relation to technical valuation assumptions. They will estimate total claims and derive risk measures such as VaR and Expected Shortfall, apply rules for premium customization, structure and evaluate contracts with flexible benefits, and implement numerical examples using appropriate computational tools.

Making judgments: Students will develop autonomy in defining and solving coverage and diversification problems in insurance portfolios, identifying the key strategic variables. They will critically evaluate the applicability of the studied models in broader contexts and under multiple sources of uncertainty, independently choosing and using mathematical and methodological tools suitable for solving complex quantitative problems in insurance.

Communication skills: Students will develop the ability to present and discuss course content, both in writing and orally. Preparing and presenting reports will enhance analytical, logical-deductive, and descriptive skills, teaching students to communicate results and arguments effectively to audiences with varying levels of expertise.

Learning skills: Students will acquire a solid foundation in actuarial mathematics, useful for approaching other quantitative courses in the Master’s program, and will develop autonomy in studying and solving problems related to risk measurement and management, also in a professional context.

The course requires basic knowledge of Financial Mathematics (interest rate; instantaneous force of interest; present value and accumulated value; term structure of interest rates and prices; portfolio immunization) and Probability Calculus (discrete and continuous random variables; Poisson, Binomial, Negative Binomial distributions, mixture; discrete-time stochastic processes).

- Introduction: deterministic financial operations; principle of financial equivalence; principle of absence of risk-free arbitrage opportunities; stochastic financial operations; sources of uncertainty and risk; principle of fairness; actuarial value; elements of an insurance contract; life insurance and non-life insurance.
- Life insurance: random lifetime; survival function; mortality and survival rates; life tables; elementary life and death insurance; demographic-financial discount factors; actuarial separability; probability distributions and actuarial values of different types of life, death, and mixed insurance; principle of contract composition; fair pure single premiums, periodic premiums, and natural premiums; mutuality, solidarity, and portfolio natural hedging; prospective and retrospective pure mathematical reserve; recurrence equations; risk premium and saving premium; first- and second-order technical bases; assessment of expected profit; tariff premium; expenses and expense loadings; full reserves; expected utility principle, risk aversion, and insurance decisions; notes on group insurance; notes on health insurance.
- Non-life insurance: insured risks; contractual conditions of compensation; random number of claims and random amount of each claim; calculation of total loss amount; calculation of the premium according to the probabilistic approach; calculation of the experience premium; loss ratio, average compensation per claim, claim frequency index, premium rate; premium customization; risk classes and merit classes; notes on bonus-malus rating systems and other forms of premium adjustment; notes on credibility theory; notes on technical reserves.
- Implicit options in insurance contracts: alterations of an insurance contract; insurance coverage with deductible and maximum guarantee limit; benefit flexibility; indexed and revaluable policies with minimum guarantee; theoretical value, intrinsic value, and maturity value of financial options; operational strategies achievable with options; arbitrage propositions; pricing constraints; binomial valuation model; risk-neutral valuation principle; notes on the Black-Scholes formula and the Monte Carlo method. (approximately 2 weeks)
- In-depth studies: scientific papers, seminar activities and/or group work related to generalizations and applications of actuarial mathematics.

Pitacco E. (2000), Matematica e tecnica attuariale delle assicurazioni sulla durata di vita, Ed. Lint (ISBN 88-8190-113-7).
Daboni L. (1993), Lezioni di tecnica attuariale delle assicurazioni contro i danni, Ed. Lint (ISBN 88-86179-03-0).
Hull J.C. (1997), Opzioni, Futures e altri derivati, Ed. Il Sole 24 Ore (ISBN 88-7187-840-X).
Other teaching materials will be communicated during the course and/or made available online by the lecturer on the E-Learning platform Moodle (course name: Matematica attuariale 2025-26_M.G. Bruno)

Dickson D.C.M. et al (2020), Actuarial Mathematics for Life Contingent Risks, Cambridge University Press (ISBN 978-1-108-47808-3).

Lessons are held entirely in person in a traditional/lecture format, integrated with seminar and/or group activities.

Attendance is not mandatory, but is strongly recommended.

The exam consists of a theoretical and practical oral test. All topics of the syllabus covered in class by the lecturer, as well as those suggested for further study and those discussed during seminar and/or group activities, will be included in the test. Alternatively, attending students may take the exam by presenting in class a written paper - possibly prepared jointly with a fellow student - under the supervision of the lecturer, analyzing and discussing a scientific paper on a topic covered by the course.
In the first case, besides the correctness of the answers, the following will contribute to the assessment: appropriate use of symbols and terminology; clarity of exposition; logical and formal rigor; mastery of theoretical reasoning and ability to generalize. In the second case, the following will be evaluated: analytical and research skills, descriptive, logical-deductive, and illustrative abilities, as well as communication and interaction skills with classmates and the lecturer.

- Illustrate the mathematical foundations useful for the valuation and immunization of a life insurance portfolio.
- Explain the theoretical assumptions underlying premium calculation in non-life insurance and illustrate the related forms of customization.
- Deconstruct and evaluate an insurance policy with a guaranteed minimum.

• Week 1:
  
  Introduction to the course; syllabus and exam
  methods; definition and fields of application of actuarial mathematics; basic deterministic
  financial operations.
  
  Stochastic financial operations; actuarial
  value; principle of fairness.
  
  Insurance policies and types of insurance; notes on group insurance.
  • Books: Pitacco


• Week 2:
  
  Life insurance: technical bases for valuation.
  
  Discrete random variable “residual life” (spot);
  life and death probabilities.
  
  Discrete random variable “residual life” (forward); annual mortality and
  survival rates.
  • Books: Pitacco


• Week 3:
  
  Life tables and obtainable information; risk
  factors and risk classes; longevity risk.
  
  Elementary insurance; demographic-financial
  separability in life and death insurance.
  
  Immediate and deferred life annuities; principle of contract
  composition.
  • Books: Pitacco


• Week 5:
  
  Periodic premiums; natural premiums; reserve
  premiums; principle of solidarity and mutuality.
  
  Prospective and retrospective mathematical
  reserve; notes on portfolio immunization.
  
  Fouret’s recurrence equation for mathematical reserves; sum at risk;
  risk premium and saving premium.
  • Books: Pitacco


• Week 4:
  
  Term life insurance; whole life insurance; mixed
  insurance.
  
  “Bespoke” policies.
  
  Fair pure single premium.
  • Books: Pitacco


• Week 6:
  
  First- and second-order technical bases;
  implicit safety loading; Homans’ contribution formula.
  
  Tariff premium; rational calculation procedure.
  
  Expense reserves.
  • Books: Pitacco


• Week 7:
  
  Explicit safety loading; expected utility
  criterion; certainty equivalent; maximum acceptable loading.
  
  Non-life insurance; types of coverage; notes on
  reinsurance coverage.
  
  Technical bases: counting distribution and severity distributions;
  assumptions.
  • Books: Daboni


• Week 8:
  
  Total amount of compensation; Maximum Probable
  Loss; fair premium and risk-adjusted premium according to probabilistic
  approach.
  
  VaR, Conditional VaR or Expected Shortfall;
  notes on reinsurance premiums.
  
  Non-life insurance and risk management; notes on actuarial coverages
  alternative to insurance.
  • Books: Daboni


• Week 9:
  
  Statistical approach to non-life premium
  calculation; experience rating; risk classification; a priori and a posteriori
  customization.
  
  Motor liability insurance: bonus-malus rating;
  multi-state approach and notes on health insurance; notes on credibility
  theory.
  
  Notes on technical reserves.
  • Books: Daboni


• Week 10:
  
  Introduction to financial options. Basic option positions; operational strategies:
  hedge, vertical spread, combination.
  
  Arbitrage propositions.
  • Books: Hull/Pitacco


• Week 11: Binomial valuation model.
  
  Principle of existence and uniqueness of
  equivalent martingale measure; notes on the Black-Scholes formula and valuation
  using the Monte Carlo method.
  
  Valuation of indexed policies with minimum guarantee; notes on surrender
  risk.
  • Books: Hull/Pitacco


• Week 12: 
  
  Seminar and/or group activities.