Investment under Risk and Uncertainty, 7.5 Credits

Contents

This course covers the mathematical and financial theory underpinning capital investments by firms. The course starts with simple investment rules, such as the net present value rule or internal rate of return rule, and then provide a formal discussion of risk aversion, utility theory, stochastic dominance, and related concepts. Following this, the Capital Asset Pricing Model (CAPM) is derived.  The course then moves from static to dynamic models, initially in discrete time and then in continuous time. In terms of methodology, the dynamic programming principle and value function iteration are the main focus. From this the real option approach is developed, which extends the net present value rule to cope effectively with features such as flexibility, timing and irreversibility, linking capital investment to the valuation of options. Numerous examples from natural resources, economics, and finance will be explored. Finally, the question of how financing (equity or debt) affects the investment process and the value of the firm, is investigated. This part sets off from the classical Modigliani-Miller theorem, but is extended within a dynamic real options context that accounts for strategic default and allows for determining an optimal capital structure.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• explain the concept of risk aversion and how utility theory and stochastic dominance implement this concept in mathematical form
• describe how the capital asset pricing model (CAPM) reflects equilibrium prices of assets in a simple static framework and how a risk-adjusted discount rate that can be used in a dynamic context can be derived from it
• mathematically formulate the real options approach in form of an optimal stopping problem that contains option like features
• explain how a firm's equity (shares) can be considered as an option on the firm's asset, and how this is reflected in the capital structure (equity vs. debt) of the firm  

Skills and abilities

• solve basic investment problems by applying the net present value and internal rate of return rule
• determine expected return of assets by applying CAPM
• compute expected utilities and certainty equivalents of risky cash flows
• solve basic dynamic problems of investment from natural resources, economics and finance, applying dynamic programming and value function iteration (in discrete time)
• apply the Hamilton-Jacobi-Bellman equation to solve relevant continuous time problems

Judgement and approach

• investigate and assess how the real option approach leads to different decision-making than the net present value approach, particularly in the presence of flexibility and irreversibility
• assess how risk-aversion changes investment behavior
• decide whether a firm should invest in a given project, accounting for different financing opportunities provided to the firm

Required Knowledge

The course requires at least 90 ECTS, including 30 ECTS in Mathematics, at least 15 ECTS in Mathematical Statistics or Statistics, and at least 7,5 ECTS in Programming Methodology, or equivalent. Proficiency in English equivalent to the level required for basic eligibility for higher studies. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

The teaching takes the form of lectures.

Examination modes

The course is assessed through written examinations in the form of two tests. From the first test students can award bonus points that can be added to the result on the mandatory final test. The bonus points are only valid on the ordinary final exam and on the first re-exam connected to the course occasion on which the bonus points were awarded. For the whole course, one of the following grades is assigned: Fail (U), Pass (G) or Pass with distinction (VG). The grade is only set once all compulsory elements have been assessed. 

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.

Examiners may decide to deviate from the modes of assessment in the course syllabus. Individual adaption of modes of assessment must give due consideration to the student's needs. The adaption of modes of assessment must remain within the framework of the intended learning outcomes in the course syllabus. Students who require an adapted examination must submit a request to the department holding the course no later than 10 days before the examination. The examiner decides on the adaption of the examination, after which the student will be notified.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications.

Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics. The course can also be included in the subject area of computational science and engineering.

In the event that the syllabus ceases to apply or undergoes major changes, students are guaranteed at least three examinations (including the regular examination opportunity) according to the regulations in the syllabus that the student was originally registered on for a period of a maximum of two years from the time that the previous syllabus ceased to apply or that the course ended.