EF 4321: Derivatives and Risk Management

Midterm Project

Fall, 2024

Instructions: The midterm project should be completed in teams. Please upload an electronic copy of your solution report in the .pdf or .docx format to Canvas – Assignments – Midterm    Project by 11:59 pm, 3 November 2024 (Sunday). Each team only needs to submit one copy. Late submissions will not be accepted.

You should only use MS Excel as the software for computation and simulations. Only answers in the .pdf or .docx solution report will be graded, so please make sure you include in the report your methodology and steps in detail, as well as formulas, screenshots, intermediate results, and other details that you deem necessary.

Question 1. Suppose you manage a mutual fund that specializes in investing in technology

stocks. You pick a portfolio of technology stocks and hope to hedge its risks using certain

index futures. The total value of your current portfolio is 100 million dollars. Please find in

the EXCEL sheet “Midterm.xlsx” the historical monthly futures price data on the S-index and N-index, as well as the historical return data on your fund portfolio (monthly return; not annualized). You can use either the S-index futures contract or the N-index futures contract to hedge the risk.

Note that the S-index futures contract has a contract unit of 50, or one S-index futures contract has a notional value of 50×[S-Index Futures Price]. The N-index futures contract has a contract unit of 20, or one N-index futures contract has a notional value of 20×[N-100

Index Futures Price]. The futures price data in the Excel spreadsheet are for indices and do not consider the multipliers of 20 and 50. Assume you can take fractions of a futures contract throughout this midterm assignment. When answering the following questions a), b), and c), assume today is the end of July 2023, and use the values given by cells B8 and C8 in the sheet “Data for Q1” when you calculate the number of contracts.

a)  Suppose you decide to use the S-index futures to hedge the risk. Should you take long or short positions in the S-index futures contracts? How many S-index futures  contracts do you need to go long (or short) in order to minimize the variance of the hedged portfolio?

b)  Suppose you decide to use the N-index futures to hedge the risk. Should you take long or short positions in the N-index futures contracts? How many N-index futures  contracts do you need to go long (or short) in order to minimize the variance of the hedged portfolio?

c)  Which is a better hedge (with the S-index futures or the N-index futures)? Explain why.

Question 2. Suppose now you are at the end of July 2023 and want to hedge the risk of your portfolio up until the end of October 2023 (for approximately three months) with the equity  index futures you picked in Question 1(c). In this question, you need to compute the varying margin balance of your futures position throughout the three-month period. For the S-index futures, the maintenance margin is $6000, and the initial margin is $6600. For the N-index futures, the maintenance margin is $7600, and the initial margin is $8360.

a) In the Excel file, you are given the futures price series for both futures from Aug 1, 2023 to Oct 31, 2023. Suppose you long/short one futures contract that you decided to use in Question 1 (whether you take a long or short position depends on your answer in Q1).

Compute the following quantities (for simplicity, omit the interest you earn in your margin account):

i. Daily Gain or Loss: the daily change in the futures price to be reflected on your account

ii. Account Balance: the margin balance after adjusting for the daily gain and loss

iii. Margin Deposit: the amount of new deposit required to meet the margin requirement

iii. New Account Balance: the amount in your margin account after placing the margin deposit

iv. Total Deposit: the total amount of capital you use to keep the position open (suppose you never withdraw money from the margin account)

v. Cumulative Profit or Loss

Question 3. In Question 2, given a particular futures price path from Aug to Oct, you computed the total deposit required to keep your futures positions alive. Let’s call this amount TD. Suppose we are on the last day of October and have closed all previous futures positions. Today, you start to go long/short one futures contract that you decided to use in Question 1 (whether you take a long or short position depends on your answer in Q1) for the upcoming 90 trading days. In this question, you will simulate 1,000 futures price paths and   get a distribution of TD. This computation helps you more precisely estimate the amount of  money required for your hedging position. In particular, this exercise allows you to answer questions like “What’s the probability that your futures position will stay alive for up to 90 trading days?” . To do this, you first need a model for the dynamics of the futures price.

a)   Suppose the daily log return on the futures price is modeled as a linear trend plus a normal random noise. That is, assuming the daily log return is

ln(Ft+1) − ln(Ft ) = μ + σ ε,

where ε is a standard normal random variable (N(0,1) distributed) and ε at different dates are independent. Use the data given in Question 2 to estimate the drift μ and the standard deviation σ of the noise term (no need to annualize the numbers for these). Report your results.

(Hint: Please compute log returns for Question 2 and estimate the sample mean and the sample standard deviation.)

b)  Given your estimated parameters, simulate the daily returns for 90 days, map each return path back to a futures price path, and compute the TD corresponding to this simulated futures price path as you did in Question 2. When doing the simulation, use the latest available futures price in Question 2 as F$ . Repeat this step 1,000 times and record the resulting 1,000 values of TD with “Excel What-if Analysis – Data Table”.

(Hint 1: A brief tutorial for Data Table that may be helpful is included in the

Midterm Project folder. For other tutorials for Data Table, there are many online resources. Read the tutorial carefully before reaching out to online sources.)

(Hint 2: You enter your position on day 0 when the futures price is F$ . You get

your first marking-to-market gain or loss on day 1, the next day. You get your last marking-to-market gain or loss on day 90, the last day.)

(Hint 3: Ifr = ln(Ft+1) − ln(Ft ), then Ft+1  = Ft  exp(r).)

c)  Now you have a distribution of the TD. You may compute a VaR type of measure of TD, i.e., what is the minimum amount of money you need to keep your futures  position alive for another 90 trading days with a 95% chance?