KTH Mathematics  

Mathematical Statistics

The investment- and risk management problems are fundamental problems that cannot be ignored. In every-day life individuals and companies are often forced to make decisions involving risks and perceived opportunities. The consequences of the decisions are affected by the outcomes of random variables that are to various degrees out of control of the decision maker. Such decision problems arise for instance in financial- and insurance markets. This course aims at presenting sound principles and useful methods for making investment- and risk management decisions in the presence of hedgeable and non-hedgeable risks.

There are two fundamental difficulties in coming up with solutions to the problems in investment and risk management. The first is that the decisions are to a large degree based on subjective probabilities of the future values of financial instruments and other quantities. Financial data are the consequences of human actions and sentiments as well as random events. It is impossible to know to what extent historical data explain the future that one is trying to model. This is in sharp contrast to card games or roulette where the probability of future outcomes to a large extent can be considered to be known.

The second fundamental difficulty is that decisions depend strongly on the attitude towards risk of the decision maker and possibly also other parties. What is your desired trade-off between risk and reward? It is difficult to accurately formalize a perceived attitude towards risk into a function that goes into the mathematical models. Mathematics can assist in translating a subjective probability distribution and an attitude towards risk and reward into a portfolio choice in a consistent way. However, uncertainty in the input to this procedure will always be passed on to the output and critical judgment cannot be replaced by mathematical sophistication.

The mathematics used to present the material is a combination of basic linear algebra, mathematical statistics and optimization theory. A student following this course is assumed to have a good understanding of the theory and tools from the above three areas, however no mathematics fancier than this is assumed or required.

Examination. There will be a written examination on Monday October 18, 2010, 14-19. Registration for the written examination via "mina sidor"/"my pages" is required. Registration requests addressed to me will not be processed.

The exam will consist of five problems. Each problem gives, if solved correctly, ten points.

Grades are set according to the quality of the written examination. Grades are given in the range A-F, where A is the best and F means failed, and Fx. Fx means that you have the right to a complementary examination (to reach the grade E). The criteria for Fx is F and that an isolated part of the course can be identified where you have shown a particular lack of knowledge and that the examination after a complementary examination on this part can be given the grade E.

Homework sets. There will be three homework sets. Note that solving the homework sets is NOT a requirement for passing the course. You may work individually or in groups of 2 persons. Each homework set gives at most 3 points which are added to your result on the October- and/or January exam (in case you do not pass any of these exams your bonus points will be lost). To obtain the points you/the group must present the solution nicely in a report which clearly shows how the problems were solved. Each report, printed on paper, must be handed in on time in order to be accepted.

Course literature This year's course literature will be the lecture notes with title "Risk and portfolio analysis: principles and methods, Part I" that can be bought at the student office of the mathematics department from the first week of the course. The material is part of ongoing work with my co-authors Henrik Hult, Ola Hammarlid and Carl-Johan Rehn. Comments and constructive critizism are appreciated.

Luenberger's book. This book, "Investment Science", was course literature in the past. It is a nice book which you may find useful. However, this year I will not use it.

Preliminary plan. Here is the preliminary plan for the course. The lectures on the dates below will not really be lectures, but rather a mix of lectures, problem demonstration, discussions of homework assignments, excercises, etc. "Examples" mean something worth looking a bit closer at, illustrating some aspects of the material presented so far.

Mo 30/813-15H1Introduction, interest rates
We 1/913-15D3Bonds, arbitrage principles
Fr 3/913-15M3Derivatives, arbitrage principles
Tu 7/9 15-17D3Examples
We 8/9 13-15M3Constructing the spot rate curve
Fr 10/9 10-12M3Quadratic hedging
Tu 14/9 15-17M3Quadratic hedging
We 15/9 13-15 D3Homework 1 presentation
Fr 17/9 10-12M3Quadratic hedging
Tu 21/9 15-17M3Convex optimization
We 22/9 13-15M3Quadratic investment
Tu 28/9 15-17M3Quadratic investment
We 29/9 13-15M3Examples
Fr 1/10 10-12 M3Risk measurement
Tu 5/10 15-17M3Homework 2 presentation
We 6/10 13-15M3Risk measurement
Fr 8/10 10-12M3Utility theory
Tu 12/10 15-17M3Utility theory
We 13/10 13-15M3Examples
Fr 15/10 10-12 M3Homework 3 presentation
Mo 18/10 14-19Exam

H1: Teknikringen 33.
M3: Brinellvägen 64, 1 tr.

Here is a map of the campus

Welcome, I hope you will enjoy the course!


To course web page

Published by: Filip Lindskog
Updated: 24/08-2010