NMAK24004U Risk Optimization

Volume 2024/2025

MSc Programme in Mathematics-Economics


Human decisions (economic and otherwise) are often made in the presence of uncertainties, that is, in advance of receiving all the necessary information. As a consequence, the outcome of a decision (e.g., returns, costs, times, losses) is typically a random quantity determined by the missing information. This exposes decision makers to risk, that is to the possibility that the outcome of their decisions is excessively undesirable.

Including measures of risk in an optimization problem offers the possibility to directly "shape" the distribution of the random outcome of interest. In turn, one gains the chance of directly controlling or minimizing risk. 

This course deals with optimizing in the presence of risk. Particularly, the course touches upon some of the classical results regarding modeling and quantifying risk and transfers these results into tractable optimization problems. These results include topics taken from utility theory, stochastic dominance, risk-reward models, chance-constraints, Value-at-Risk, Conditional Value-at-Risk, coherent measures of risk, time-consistent measures of risk, deviation measures.

An exposition of the central theoretical results will be followed by practical project work on case studies inspired by real-life problems.

Learning Outcome


  • Various ways of measuring risk
  • Relationships between risk measures
  • Characteristics of different risk measures
  • General formulations of optimization models that minimize or constrain risk



  • Describe alternative ways of measuring risk
  • Describe their advantages and disadvantages and relationships to other measures
  • Formulate optimization problems that handle risk
  • Solve practical risk optimization problems and describe their outcome



  • Recognize and structure a decision problem affected by uncertainty and propose a suitable way for managing risk
  • Choose a suitable measure of risk for the decision problem at hand
  • Implement and solve risk optimization problems
  • Deliver and explain solutions of risk optimization models
  • Compare the results of alternative risk optimization models

Lecture notes.

A basic course in probability theory. In addition, it is useful to have some experience with optimization comparable to, e.g., Operationsanalyse 1 or Applied Operations Research.

Academic qualifications equivalent to a BSc degree is recommended.
Lectures (approximately 18 hours) for the first part of the course (3-4 weeks). Project work (individual or in groups) on an assignment for the second part of the course.
  • Category
  • Hours
  • Lectures
  • 18
  • Preparation
  • 40
  • Project work
  • 100
  • Guidance
  • 7
  • Exam Preparation
  • 40
  • Exam
  • 1
  • Total
  • 206
Feedback by final exam (In addition to the grade)
7,5 ECTS
Type of assessment
Oral exam on basis of previous submission, 30 minutes (no preparation)
Type of assessment details
The students must hand in a project report that will form the basis of the oral exam.

The oral exam starts with the student presenting and discussing their work in the project (10-15 min) and continues with questions on the contents of the course (ca. 10 min).
Without aids
Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners

Same as the ordinary exam.

The project must be delivered to the course responsible at least two weeks before the date of the reexam.

Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under intended learning outcome.