MTH202

Introduction to Financial Mathematics

AUTUMN SEMESTER 2025/26

Individual Assignment ONE

SUBMISSION DEADLINE: 8:00 PM on Sunday November 16, 2025

Introduction

This part of the course assessment is worth 15% of the final mark for the course, and consists of a take-home course assignment that will be worked on and submitted individually.

This project aims to practice your skills in hedging market risk by constructing  a portfolio involving futures. The relevant information about the securities prices can be founded through Yahoo Finance   (http://finance.yahoo.com), Bloomberg  terminal (accessible at IBSS), Wharton Research Data Services (register via the university email account at https://wrds-www.wharton.upenn.edu/), or WIND terminal  (accessible at Library). In this assignment, you were provided with the following scenario.

Scenario:

Suppose you are working on a fund of $20 million which invests in a portfolio of the S&P500 component stocks, e.g., Apple, HP, BP and etc. This portfolio has been constructed since November 1st, 2024. On May 30, 2025, you were worried about the Trump tariffs and decided to set up a hedging strategy to protect your fund by using the S&P500 index futures for the following four months, e.g., June 1, 2025 to October 10, 2025,  and so the trading position in the futures contract would be closed on October 10, 2025.

You are required to construct an optimal hedging position in the S&P500 index futures, and analyze its performance during the period from June 1, 2025 to October 10, 2025.

1) All the securities in your portfolio are selected from the component stocks of the S&P500 index, which was constructed on November 1st, 2024.

2) The securities should be selected from different industries, e.g., bank industry, media industry, and manufacturing industry etc.

3) The number of stocks in your portfolio must be between 12 and 20, e.g., 12 ≤ n ≤ 20.

4) The daily returns of all the securities in your portfolio are needed, e.g., a time series of daily returns from November 1, 2024 to October 10, 2025.

5) The daily risk-free rates (e.g., 1-month T-bill rates in US) for the same period are needed.

6) The S&P500 index futures that matures on December 2025 (E-mini S&P 500 Futures, ESZ25) is used for the hedging purpose and the time series of daily close prices from November  1, 2024 to October  10,  2025 is needed.  Note that each futures contract is for delivery of $50 times the index.

7) The dividend yield on the S&P500 index is 1% per annum.

Requirements:

I.     Given that the securities in your portfolio are only individual stocks (namely, excluding the equity index), you apply the Capital Asset Pricing Model (CAPM) to complete the following tasks (20%):

a) Provide the descriptive statistics of the individual stocks, your portfolio and the market portfolio (S&P500 index) in terms of excess return, standard deviation and covariance. Plot the time series of each of these (daily) returns.

b) Estimate the beta for each individual stock in the portfolio.

c)  Estimator the beta for your portfolio.

d) Plot all the betas in b) and c) in the space of (X=beta, Y=expected return) to confirm the Security Market Line (SML).  Note that the beta of the market portfolio is equal to ONE.

e) You should report and summarize these figures and results in the report.

II.    Suppose that on May 30, 2025, you develop a hedging strategy for your portfolio constructed in I) by using the S&P500 index futures that matures on December 2025, and close your futures trading position on October 10, 2025. You then apply the optimal-hedging-ratio approach to complete the following tasks (60%):

a) Develop the hedging strategy with the minimal variance hedging ratio which is estimated based on the time series of price changes between the portfolio and the index futures.

b) Develop the hedging strategy with the optimal hedging ratio which is estimated using the CAPM model;

c)  Suppose that the S&P500 index and the close price of the December  index futures are 5911.69 and 6021.75 on May 30, 2025, respectively. Analyze the total value of hedging position on October 10, 2025 by following the strategies in

a) and b), including the gain/loss on the hedge when the S&P500 index and index futures fall into the following scenarios:

d) Analyze the dynamics of the gains/losses of your trading positions in the index futures by following the strategies in a) and b) during the period from June 1, 2025 to October 10, 2025, given the real market data you have collected.

e) Analyze the dynamics of the total value of hedging position by following the strategies in a) and b) during the period from June from June 1, 2025 to October 10, 2025, given the real market data you have collected.

Note that to solve this part, you need carefully check out Q2, 3&4 in tutorial 5.

III.  Assess the advantages and disadvantages of the optimal hedging strategies developed in II). For example, you may compare the statistics of both the profits/losses generated by these two strategies and the dynamics of the total value of the hedged portfolio (e.g., your stock portfolio plus the hedging position in the index futures) in II). You may further analyze the potential risks when they are used in various market conditions (bullish markets vs bearish markets) (with the maximum of 600 words). (20%)

Note: There are three ideas for your reference:

1) Comment the difference in the construction of two hedging strategies (one with minimal variance and the other estimated from beta).

2) On the one hand, on May 30, 2025, you made the expectation about the terminal value of your hedged portfolio across a range of scenarios on October 10, 2025. This is just your expectation based on scenario analysis. On the other hand, you do know the terminal value of the hedged portfolio on October 10, 2025, given the real data. So, there  is a gap between your expectation and real results you  get on October 10, 2025. Make comments on this gap.

3) During the period from May 30 to October 10, 2025, your hedging strategies may present different performances. For example, in the bearish market (e.g., market declines), which one performs better? In the bullish (e.g. the market grows), which one  performs better?  Remember that the hedging  strategies are used to provide protections on your portfolio, e.g., to offset the loss of your portfolio in  the bearish market, and to minimize the impact on your portfolio in the bullish market.

Individual Assignment Guideline

This individual assignment assesses Learning Outcome A-C.

In this assignment, you need:

1) Show all the results with comprehensive interpretations in a report (with the maximum 20 pages).  The writing should follow the  academic style. of leading finance journals (e.g., JF, JFE, and RFS).

2) You should use programming software (e.g., MATLAB, Python, R, STATA, etc.) to finish this assignment.

3) Submit relevant and supportive files (e.g., programming code script and data) in Learning Mall. For instance, plots in I) and the construction process of the required hedging strategies in II).

4) Prepare Tables and Figures in academic style.

Formatting requirement:

1) The assignment should follow APA reference style.

2) Tables and figures should be able to stand alone. Make column headings descriptive and easily understood. Design all variables and abbreviations.

3)  Number tables, figures, and equations with Arabic numbers.  Enclose equation numbers in parentheses and place them in the right margin.

4) On the title page, include the title, student’s name, student’s ID. Double- space all text, including abstract, footnotes, and references.

As the outcome from your individual project, you are expected to submit a report. Please disclose the detailed process/results as much as possible (e.g., the statistics of those portfolios and the data sets to support your plots).

The deadline of the assignment submission to Learning Mall is at 8pm on November 16, 2025 (Week 9). You may download all the required financial information via Yahoo Finance   (http://finance.yahoo.com),    Bloomberg terminal (accessible at IBSS), Wharton Research Data Services (register via the university email account at https://wrds-www.wharton.upenn.edu/),  or WIND terminal (accessible at Library).

Your analysis has to be your own, demonstrating your own ideas  and independent and critical thinking (and not those of somebody else).

However, your report must include:

1) A brief description of the project

In the first section, your work should contain a formal introductory section that provides an overview of the project, including the title of the project, the data collection and preparation (especially for the data source), securities selection, and the main goals that this project aims to achieve.

2) Process of the optimal hedging strategies and performance analysis

After preparing all the required financial data, you are ready to complete the tasks in I) and II) which can be presented in two separated sections. It is suggested that you describe the process of portfolio construction in detail and report their statistics accordingly, and then provide the details about the development of the optimal hedging strategies. Also, please clearly explain how you obtain the required results, supported by your models in programming files.

Note: the lecture and tutorial in Weeks 5-7 will demonstrate the strategy how to complete these tasks in a simplified case.

3) Comments on results

In the final section, you need evaluate the performance of the optimal hedging strategies against distinctive market scenarios in   the asset management process.  It is very important for you to understand  the advantages/disadvantages of each method in  specific market conditions, which will help to improve your skills in investment management in practice.

Declaration of generative AI in scientific writing

Students must declare the use of generative AI in report writing upon submission of the paper. The technology must be applied with human oversight and control and authors should carefully review and edit the result, as AI can generate authoritative-sounding output that can be incorrect, incomplete  or biased. Students are ultimately responsible and accountable for the contents of the work.

The use of generative AI and AI-assisted technologies in report writing must be declared by adding a statement at the end of the manuscript when the report is submitted.  The statement should be placed in a new section before the references list. An example:

Title of new section: Declaration of generative AI and AI-assisted technologies in the report writing process.

Statement: During the preparation of this work the author(s) used [NAME TOOL / SERVICE] in order to [REASON]. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the written report.

The declaration does not apply to the use of basic tools, such as tools used to check grammar, spelling and references. If you have nothing to disclose, you do not need to add a statement.

AI Use Policy: Declaring your use of AI for this assignment is optional and will earn a 10-mark bonus.  However, submitting AI-generated work without a declaration will result in a 60-mark penalty.

Submission: each student submits ONE report and TWO supporting files (data and programming code script).  All the other forms can be submitted to the tutors/module leader privately.

Suggested references for academic writing:

Atilgan, Y., Bali, T. G., Demirtas, K. O., & Gunaydin, A. D. (2020). Left-tail momentum: Underreaction to  bad  news,  costly  arbitrage  and equity  returns. Journal of Financial Economics, 135(3), 725-753. (See Table 1 for summary statistics and Section 3 for how to discuss empirical results).

Geng, Z., & Pan, J. (2024). The SOE premium and government support in China's credit market. Journal of Finance, 79(5), 3041-3103. (See Table 1 for summary statistics and the display of figures).