Linear Algebra and Calculus.
This course introduces the students to mathematical models and computational
methods for static and dynamic optimization problems occurring in insurance and finance.
We shall discuss linear and non-linear optimization models of finance, dynamic (sequential)
optimization, optimization under uncertainty, mathematical models of
risk and their application. Additionally, duality theory and its use in insurance
and finance will be stressed. The students will be familiarized with concept of
risk and risk-aversion. The models involve knowledge of probability, optimality
conditions, duality, and basic numerical methods. Special attention will be paid
to portfolio optimization, to risk management problems and economic scenario generator able to generate coherent and market consistent scenarios for a variety of asset classes.
At the end of the course the student will be able to:
1. Formulate optimization problems associated with various problems in insurance and
finance such as dedication problems, immunized bond portfolio model,
portfolio optimization using mean-variance models or coherent measures
of risk and/or risk constraints, tracking an index, etc.
2. Understand the concept of risk and be able to formulate and apply several
mathematical models of risk based on utility functions, coherent measures
of risk, and risk-constraints.
3. Calculate the efficient frontier determined by a mean-risk model; use the
one-fund and two-fund theorems.
4. Use the concept of stochastic orders, be aware of their relation to risk measures
and utility functions.
5. Formulate finite-horizon dynamic optimization problems based on Markov
and non- Markov discrete time processes.
6. Apply stochastic optimization methods for option pricing and for asset/liability
7. Understand the Economic Scenario Generator to generate coherent and market consistent scenarios.
8. Implement mathematical optimization models for ALM and PFM in the AMPL environment.
9. Assess the solution of the implemented models and interpret the results in a decision-making perspective.
The course will discuss and present the methods and techniques that are relevant for evaluation and modelling the intertemporal risk in finance and insurance.
Specifically, the course will cover the following topics:
- Economic Scenario Generator (ESG).
- Static and Dynamic Risk measures.
- Dynamic Programming for multi period investment problems.
- Multistage Stochastic Programming for multi period investment problems.
- Asset Liability Management (ALM) using dynamic programming and two-stage/multistage stochastic programming.
- Pension fund management (PFM) using dynamic programming and two-stage/multistage stochastic programming (defined benefit and defined contribution).
- Individual Retirement Pension via stochastic programming.
The course consists in traditional theoretical lectures and practical lab sessions (using AMPL and MATLAB software). The emphasis will be on the practical implementation of the models using AMPL and scenario generation using MATLAB software. Both traditional lectures and practical sessions aim at fostering participation and class discussion.
The exam consists in two parts:
- Oral discussion about applied assignments and case studies (50% of the final grade).
Students may work in small groups or individually.
- Final oral exam (50% of the final grade).
The course material will be provided by means of the e-learning platform of the University of Bergamo.
If the teaching activity will be mixed or in remote mode, changes can be done compared to what stated in the syllabus, to make the course and the exams available also in these modalities.
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