# Coursebooks 2017-2018

## Optimization methods

Pegoraro Fulvio

English

#### Remarque

For sem. MA1. Special schedule: see the IF website http://sfi.epfl.ch/mfe/study-plan

#### Summary

This course presents the problem of static optimization, with and without (equality and inequality) constraints, both from the theoretical (optimality conditions) and methodological (algorithms) point of view. Economics and financial applications will be provided.

#### Content

Topic 1 - Static optimization

Optimization programs with and without constraints. Examples of optimization problems in economic theory. Existence of solutions. Unconstrained optima: first-order and second-order conditions. Equality constraints and the Theorem of Lagrange. The Constraint Qualification. The Lagrange multipliers and their interpretation. Using the Theorem of Lagrange. Two examples from economics: Consumer and Producer Theory. One example from finance: optimal portfolios
and mean-variance analysis. Inequality constraints and the Theorem of Kuhn-Tucker. The Constraint Qualification. The Kuhn-Tucker multipliers. Using the Theorem of Kuhn-Tucker. An illustration from Consumer and Producer Theory. The general case: mixed constraints. Algorithms for univariate and multivariate nonlinear optimization (binary search, golden section search, Newton's method, steepest descent).

Topic 2 - Application to the choice under uncertainty

The investor's risk attitudes. Mean-variance criterium and expected utility criterium. Risk premium and certainty equivalent. Arrow--Pratt coefficient of risk aversion and corresponding utility functions. The investor's optimal portfolio with one risky asset. Optimal portfolio and wealth effect. Optimal portfolio with multiple risky assets. Portfolio expected return and variance. Portfolio diversification. The impact of different risk attitudes in portfolio choice.

Topic 3 - Application to the Estimation of the term structure of interest rates

The term structure of interest rates. Forward rates and the forward term structure of interest rates. Interpolation techniques and estimation methods: Ordinary Least Squares (OLS) vs Nonlinear Least Squares.

#### Keywords

Optimization program, equality and indequality constraints, Lagrange and Kuhn-Tucker theorems, algorithms, choice under uncertainty, mean-variance and expected utility criteria, term structure of interest rates estimation.

#### Learning Prerequisites

##### Important concepts to start the course

Basic concepts of linear algebra, mathematical analysis and probability.

#### Learning Outcomes

By the end of the course, the student must be able to:
• Describe optimization programs with and without equality or inequality constraints
• Solve optimization programs with and without equality or inequality constraints
• Describe algorithms adopted to solve such a univariate and multivariate optimization problems.
• Discuss the expected utility criterium and the mean-variance criterium.
• Present how the different risk attitudes of the investor can be characterized via the shape of the utility function.
• Discuss the investor's optimal portfolio, the impact of risk aversion and the wealth effect.
• Present the problem of interpolation and estimation of the term structure of interest rates.
• Discuss the problem of optimal portfolio in a static market model with a finite number of assets.

#### Transversal skills

• Use a work methodology appropriate to the task.
• Set objectives and design an action plan to reach those objectives.
• Demonstrate the capacity for critical thinking
• Use both general and domain specific IT resources and tools

#### Teaching methods

Slides. There is no required book for the course.

#### Assessment methods

The grading will be based on exercises (30%), and (70%) final exam. The final exam is closed-books and closed-notes.

#### Resources

No

##### Bibliography

- Dixit,, A. K., "Optimization in economic theory", Oxford University Press, second edition.

- LeRoy, S. F., and J. Werner, "Principles of Financial Economics", Cambridge University Press.

- Luenberger, D G., "Linear and Nonlinear Programming", Kluwer Academic Publisher, second edition.

- Claus Munk, "Financial Asset Pricing Theory", Oxford University Press.

- W. Rudin, "Principles of Mathematical Analysis", McGraw-Hill Education, third edition.

- C. P. Simon and L.E. Blume, "Mathematics for Economists", W. W. Norton and Company.

- R. K. Sundaram, "A First Course in Optimization Theory", Cambridge University Press.

##### Notes/Handbook

Slides for each lectures will be provided.

### Reference week

MoTuWeThFr
8-9
9-10EXTRANEF_126
10-11
11-12
12-13
13-14
14-15
15-16 EXTRANEF_126
16-17
17-18
18-19
19-20
20-21
21-22

Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German