MATH 4/5779 – Math Clinic

 Mathematical Models for Risk Analysis of Portfolio Investments

Sponsored by K. David Jamison - actuary, Watson&Wyatt


Spring Semester ‑ 2003

Professor: Weldon A. Lodwick


Office: CU-Denver Building, Room 622

Telephone: 556‑8462 (office - voice mail), 556‑8442 (secretary), 556-8550 (fax)


Web Site:


Office Hours:   Tu        5:25 –   6:55 PM            CU-Denver Bldg 622

W         9:00 – 10:00 AM            CU-Denver Bldg 622

Th        2:55 –   3:55 PM            CU-Denver Bldg 622

                        Other times by appointment


Text: I will suggest several texts and make some of these available to be checked.  Periodically you will also receive several packets of articles and/or notes.  Thus, for the moment, there will be no required text.  Note: Professor H. Greenberg will be giving a GAMS tutorial (see every Thursday from 5:30PM to 6:45PM in room 641 of CU-Denver Building (see It is free and you are welcome/encouraged to attend. GAMS is a professional mathematical programming and modeling software system.


Students with Disabilities: If you have a disability that requires accommodation in this course, please see me as soon as possible.  I am happy to make appropriate accommodations provided timely notice is received.


Cell Phones: You are to turn off your cell phones prior to entering class.

Objectives of this Mathematics Clinic: The clinic is primarily a pedagogical tool where one learns applied mathematics by solving problems faced by the sponsoring institution.  Working in research teams to develop results associated with a project (solving a set of problems and presenting the results) is an integral part of every clinic.  Thus, we will try to solve problems that are of current concern.  In particular, for this semester, our objectives will be:

1.       To develop a good prototype asset computer model for risk management of portfolio investment of retirees.

2.       To incorporate the upper/lower probability into the asset model to enable the computation of the distribution of risk of wealth for an individual retiree for each retirement year and compare it to existing methods (Monte Carlo for example).

3.       To develop an optimal investment strategy and add it to the expanded asset model with upper/lower probability capability.

4.    To incorporate a broader range of uncertainty into the models - these methods come out of recent research



The proposed outline is the initial guess of the topics that will be fruitful to investigate.  Research is a process of discovery when one does not know, so the rule is that we will modify our topics during the semester.  Thus the proposed outline will undoubtedly change as we learn more during the semester.


This semester we will study traditional and contemporary mathematical models for risk analysis of portfolio investments of pension funds.  The objective will be to develop and test prototype algorithms for the analysis of portfolio investment risk.  The tentative topics we will cover are:

I.                    Theory

A.      Portfolio Models – "Efficient Asset Management" by Michaud, articles

B.      Utility and Stochastic Dominance Theory – Overview

C.      Monte Carlo Methods – "Efficient Asset Management" by Michaud, articles

D.     Upper and Lower Probability Methods – R.E. Moore, Williamson&Downs, Jamison & Lodwick, Lodwick & Jamison, Berleant, and Ferson articles

E.      Clouds – Neumaier articles

F.      Fuzzy Set Methods – Tanaka, et. al., Inuiguchi, et. al.

II.                 Algorithm Development and Testing

A.      Nonlinear Programming and Simulation – Markowitz, Michaud

B.      Upper and Lower Probabilities – Jamison & Lodwick

C.      Clouds – Neumaier

D.     Fuzzy Set Methods – Tanaka, et. al., Inuiguchi, et. al.


The work this semester will be divided as follows:

I.           Introduction to the problem, models, approaches and algorithms (first 4 weeks)

II.                 Development of a good working prototype asset management model (second 4 weeks)

III.               Incorporate into the asset model, upper/lower bound approach of J&L (third 4 weeks)

IV.               Develop and incorporate an optimal investment strategy component into the asset model with upper/lower bound approach (fourth 4 weeks)

V.                  If there is time, incorporate uncertainty into the asset model with upper/lower bounds for optimal investment



There will be three projects:

  1. Project 1: To development of a working, prototype asset management model software.
  2. Project 2: To incorporate into the prototype asset management model software, upper/lower probability/possibility bound capability.
  3. Project 3: To add an optimal investment strategy component to the expanded asset management model software containing upper/lower bound capability.


The class will be a team that will be divided into several groups.  All groups will be working on subtasks leading to the completion of projects 1, 2 and 3. That is, each group will have responsibilities associated with all projects.  Each project will be subdivided into tasks and assigned to the groups.  Each group in turn will divide the tasks into subtasks and assign these to individuals within the group.  Software development involves research to create the software, the creation of the software, the testing and analysis of the software and the documentation.



I believe that teaching is a process that involves an active partnership.  My role is that of a guide to your learning.  Therefore, I am responsible to open the way, to encourage, and to nudge you toward your own learning.  In the context of the math clinic, I will try to model the process of applying mathematics to the risk analysis of portfolio investment problem. I will help guide you toward this learning by providing mathematics for you to experience.  It is my aim to communicate mathematics in a way that is supportive and nurturing of your efforts. Your role is to find a way to experience and articulate the mathematics that is presented and that you encounter.  I believe that it is your responsibility to let me know when you find yourself not understanding mathematical concepts that are presented in class.  Once you make this known, it is our responsibility to work on trying to attain clarity.  I will try to be as proactive as possible.  I believe that results on projects give us the opportunity to clearly see where the areas of mathematical understanding are and what areas need more attention.



By the end of the semester you should be able to read, understand and apply appropriate methods associated with aspects of risk analysis of portfolio investments for pension plans we’ve studied this semester to correctly solve associated problems.  Secondly, given a problem in the area of risk analysis of portfolios that we have studied this semester, you should be able to: (i) translate the description of the problem into an algorithm, (ii) choose and apply the appropriate software method(s), (iii) obtain the correct solution(s), and (iv) (correctly) interpret and display results.  Lastly, by the end of the semester you should be able to judge, for yourself, the veracity of statements made in the areas of our study.



Each person on a team will execute a project (identify a set of problems, find solution methods, present the results and write-up the results).  In particular, the following are components that will be evaluated.

  1. Participation – attendance and contributing to class interactions/discussions (10%)
  2. Annotated bibliography – First quarter report (10%)
  3. Projects 1, 2 and 3 – Second, third and fourth quarter reports (60% total)
    1. Write-up (5% each)
    2. Software, analysis, testing (15% each)
  4. Final Presentation (10%)
  5. Final written report (10%)

** Graduate students will have extended content and be held to higher standards.

The grade assignments are on the 10 percent scale (A = 90%-100%, B = 80%-89%, C = 70%-79%, D = 60-69%).



Group/team selection – on or before January 31st

First quarter reports and annotated bibliography – February 18th

Project 1: task identification – February 20th

Project 1: division of labor for each group – February 21st

Project 2: task identification – March 20th

Project 2: division of labor for each group – March 21st

Second quarter reports (project 1) – March 21st

Project 3: task identification – April 17th

Project 3: division of labor for each group – April 18th

Third quarter reports (project 2) – April 22nd

Fourth quarter reports (project 3) – May 9th

Final Presentation – May 13th and May 15th

Final reports – May 16th



General advice: Keep all materials that I turn back in case you think I have not credited you with the points you earned.  I can only correct your score if you have what I have turned back to you. It is a good idea to copy anything that you turn in just in case I lose what you turn in.  Please check to make sure that the points you earned are the points I have recorded.  Note: The statistics that I have read about correctness of professors in recording grades state that there is a 6% error rate in our recording of your grades.  Please make sure that I have correctly recorded your points.



Adds, drops and incomplete grades: See Schedule of Courses for the relevant dates with respect to adding and dropping this course.  Given the budget cuts facing universities, you must be registered by the dated specified or you will not get credit.  The incomplete policy of the Mathematics Department and the College of Liberal Arts and Sciences is strictly enforced.  Incomplete grades are given only in situations in which a student who has been in good standing all semester, is prevented from completing a course assignment (for example the final exam) due to circumstances beyond her/his control (for example, hospitalization, jury duty, revised job assignments, death in the family).


Legitimate Excuses: Legitimate excuses are for reasons that are beyond your control.  You may be required to produce an official, signed excuse.  If you are needed in a wedding, for example, you must talk to me prior to the (blessed) event.  If you are legally arrested, then this is not a legitimate excuse.  For matters that are within your control, the general rule is that it is not excused.  However, talk to me prior to the event.


Teams: If all items of the "Division of Labor" are correctly fulfilled by the responsible person(s), then all members of the team will receive the same point distribution.  An individual in a team will be rated differently for one or more of the following reasons:

·         The individual's share of the labor as outlined in the "Division of Labor" is not fulfilled

·         The individual's portion is incomplete

·         The individual's part is poorly completed

·         The individual failed to meet with the team to plan and carry out the project





INSTRUCTIONS FOR PROJECTS: A project consists of:

1.       Proposal – Each of the three projects will be divided into tasks and assigned to each group so that the assignment is equitable.  These tasks and assignments need to be written up and submitted to me.  Once the tasks have been identified, assigned and approved, a division of labor is written by each of the groups.

2.       Division of labor – Each group must take their tasks and subdivide them into subtasks that are assigned to individuals in the group with an associated due-date.  A division of labor is a formal contract between the members of the group.  Once the tasks have been approved and a written division of labor submitted, the group needs to schedule of meeting with me so that we can go over the division of labor, its associated responsibilities and expectations.

3.       Software

a.       Code - the actual computer implementation of the project.  Attention must be paid to efficiency, readability and portability.

b.       User interface – the way information is passed to the software must be compelling to the client.

c.       Data and inputs

d.       Execution - the algorithm as run must correctly perform what it was designed to do.

e.       Output - relevant, clear display of solution (tables, graphs, images).

f.    Ease – ease of use.

g.    Documentation – an in-line and hardcopy of the documentation on how to use the software needs to be written.  Moreover, help files must be part of the software.

4.       Testing and analysis

a.       Testing - this part in the context of our clinic consists of running the software developed on the test problems and comparing results to Monte Carlo simulations of the same set of problems.  We will be compiling a set of test problems as a part of our clinic.

b.        Analysis - the purpose of an analysis is to get you to critically evaluate the results obtained from the software as it was run on the test problems.  Part of an analysis is a critique of the software.

5.       The Clinic Report – Each team will need to be responsible for parts of the final clinic report that will be delivered to our sponsor and is a part of the mathematics department’s published Clinic Report Series.  This will be done in MS-Word or Latex.  The final report will (subject to modifications we uncover) consist of:

a.       Introduction – clinic director

b.       Project 1

                                                              i.      Theoretical foundations – theory, application, algorithms

                                                            ii.      Software – description

                                                          iii.      Results – conclusions, limitations and improvements

c.       Project 2 (same as project 1)

d.       Project 3 (same as project 1)

e.       Opportunities for further research

f.        Conclusions

g.       Bibliography

h.      Appendices (Source code, test problems, documentation)