SYLLABUS

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)

**E-Mail:** weldon.lodwick@cudenver.edu

**Web Site:**
http://www-math.cudenver.edu/~wlodwick

** **

**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
http://www.gams.com/docs/document.htm) every Thursday from 5:30PM to 6:45PM in
room 641 of CU-Denver Building (see
http://carbon.cudenver.edu/~hgreenbe/courses/S03/Readings.html). It is free and
you are welcome/encouraged to attend.

** **

**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.

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

**Projects:**

There
will be three projects:

**Project 1:**To development of a working, prototype asset management model software.**Project 2:**To incorporate into the prototype asset management model software, upper/lower probability/possibility bound capability.**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.

**MY APPROACH TO TEACHING**

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.

**OUTCOMES**

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 weve 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.

**EVALUATION**

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

** **

**IMPORTANT DATES:**

Group/team selection on or
before January 31^{st}

First quarter reports and
annotated bibliography February 18^{th}

Project 1: task
identification February 20^{th}

Project 1: division of labor
for each group February 21^{st}

Project 2: task
identification March 20^{th}

Project 2: division of labor
for each group March 21^{st}

Second quarter reports
(project 1) March 21^{st}

Project 3: task
identification April 17^{th}

Project 3: division of labor
for each group April 18^{th}

Third quarter reports
(project 2) April 22^{nd}

Fourth quarter reports
(project 3) May 9^{th}

Final Presentation May 13^{th}
and May 15^{th}

Final reports May 16^{th}

** **

**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.

**POLICIES**

**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 departments
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)**