Instructions

This is an individual assignment, so you must complete it on your own without any external assistance. Failure to comply with this requirement will constitute a departure from academic integrity and will be dealt with accordingly.

As I mentioned in class, before tackling this assignment, you should: 1) read your lecture slides thoroughly, 2) solve all the problems included therein and, 3) complete the relevant problems in the individual problem set.  This assignment focuses explicitly on the mate- rial that we covered in class, so you should steer clear of external sources of information (textbooks, internet, etc.) to avoid confusion and to save time.

While you may use spreadsheets to produce your solutions, please, present your as- signment paper in one single document constructed with a word processor, using Cambria 12-point font, or equivalent, and submit your document in PDF to the course portal’s drop box. Non-PDF submissions will be rejected by the system. Your submission should be con- cise, easy-to-read, visually appealing, and should abide by the page limit that is specified for each question.  You should present your answers in neatly-constructed tables whenever feasible, with no more than one table per sub-question.  If you wish to scan handwritten answers into your document, make sure that your handwriting is sufficiently neat and that it meets the above minimal font size requirement.

To enable me to grant you part marks, should your final answers be incorrect, indicate how you performed your key calculations in a brief note below each question.  When an explanation is required, please be clear and concise. I mark the assignment one question at a time to ensure that each question is graded consistently across all submissions. To facilitate my grading process, please identify yourself by name and student ID on the cover page of your paper and print your student ID on the upper right-hand side corner of each page.

Please, submit your assignment paper on time, to avoid the steep late-submission penalty, and abide by the formatting and presentation requirements stated above, to avoid the asso- ciated penalties.

Best wishes.

Prof. Gagnon

Question 1: (Maximum 2 pages)

Today is January 20, 2023, and the current term structure of Canadian money market interest rates (expressed in decimals) is as follows:

(A) If you invest $1 in a money market account today, how much money will you have in your account at the end of each period included in the above table?

(B) Calculate the 3x6, 6x9, 9x12, 12x15, 15x18, 18x21, and 21x24 interest rates implied by the above interest rates.

(C) If you invest $1 today at the 0x3, roll over your balance at the 3x6 for the next three months, and repeat this process until January 18, 2025, how much money will you have in your account at the end of each period?

(D) Construct the term structure of semi-annually compounded interest rates that are equivalent to the interest rates displayed in the above table.

(E) One of your clients wants to fix her borrowing rate with a forward contract for a‘one-

year’ $1 million loan starting on January 20, 2024, and ending on January 18, 2025. Calculate the interest rate that you will be able to offer her in this contract and spell out the components of the forward forward transaction that you will need to execute to unwind the interest rate risk exposure that you will acquire from her in the process.

Question 2: (Maximum 2 pages)

You are a derivatives market-maker based in Switzerland providing risk management services to various institutional clients based in Switzerland, Germany, and the U.K. Today is January 20, 2023, and the term structure of money market interest rates (expressed in decimals) is as follows:

(A) One of your clients wishes to enter into a forward contract expiring on January 18,

2025, to purchase one thousand units of an asset that is trading for CHF 1,500 in the Swiss market today. Itemize the steps of the manufacturing process that you will need to implement to serve this client, in the proper sequence, and determine the minimum price at which you will be willing to enter into this contract. In a few words, explain why the spot and the forward prices are different.

(B) Upon further analysis, the asset will pay CHF 150 in income on January 20, 2024.

In view of this, itemize the steps of the manufacturing process that you will need to implement to serve this client, in the proper sequence, and determine the minimum price at which you will be willing to enter into this contract. In a few words, describe the impact of an asset’s income yield on the asset’s price in the forward market relative to the spot market.

(C) As it turns out, the asset is trading in the forward market for CHF  1,200 today, with the same expiry date as the contract that you are constructing for your client. Compare this price to your manufacturing cost and determine whether the disparity between the two prices represents an arbitrage opportunity.  If so, itemize the steps that an arbitrageur will need to implement to exploit this price disparity, in the proper sequence, and calculate the profit that she will be able to realize from this process.

Question 3: (Maximum 2 pages)

You hold a USD 10 million notional position in the 4 1/8%, 11/15/32, U.S. Treasury note and you expect the Federal Reserve Board to push the overnight interest rate up further at its next meeting. In view of this anticipated policy action, you have decided to hedge your interest rate risk exposure with U.S. Treasury futures contracts, which trade on the CME Group’s exchange. You are considering two candiates for this purpose, namely the 10-year U.S. Treasury note and the long bond futures contracts. The relevant details regarding your spot position and the two futures contracts are provided in the table below.

(A) Using the DV01 methodology, determine the futures position that will be required to mitigate your interest risk exposure.  The DV01s in the table are for the notional amounts shown in the second row. Please, spell out your calculations carefully.

(B) Using the duration-based methodology, determine the futures position that will be required to mitigate your interest risk exposure.  Please, spell out your calculations carefully.  Briefly comment on the difference between the futures positions under the DV01 and the duration-based methodologies.

(C) Using the Bloomberg dataset (see spreadsheet), assess the in-sample hedging perfor- mance of the duration-based methodology for each contract.   In a table similar to the one shown on slide #29 of presentation #6, report the descriptive statistics for the time-series of the daily P&L for the unhedged and the hedged position for each contract, i.e., mean, minimum, maximum, standard deviation, and number of observa- tions. For this purpose, exclude all the observations for which one or more data points are missing.

(D) Plot the time-series of the in-sample P&L for the unhedged and the duration-based hedged position for each contract and pick the contract that exhibits the highest hedg- ing performance. Discuss briefly.

Question 4: (Maximum 2 pages)

One of your clients holds a physical position of 1,000 troy ounces of gold to allay her concerns about the runaway increase in the consumer price index over the past several months and the recent disruptions experienced by the global supply chain. In view of the recent surge in the price of gold, she is concerned that the precious metal is fast approaching its next technical peak so, ironically, she would like to hedge her gold price risk exposure for the next little while.  Consequently, you are advising your client to use the gold futures contract trading on the CMX-Commodity Exchange (COMEX), which is a division of the CME Group. To assess the hedging effectiveness of this futures contract, you have tasked your assistant to assemble a daily time-series dataset consisting of gold spot and futures prices. The futures price series is constructed by Bloomberg by stringing together the settlement prices of a sequence of nearby contracts.  The sample period starts on January 20, 2021, and ends on January 20, 2023.

(A) The first step in your process is to clean up this dataset by removing the observations for which one or more data points are missing.  The second step is to produce daily continuously compounded return series, Ln(Pt /Pt − 1 ), from the spot and futures price series. To ensure that we are all on the same page, present descriptive statistics (mean, standard deviation, minimum, p5, p10, p25, p50 (median), p75, p90, p95, maximum, and total number of observations) for the two price series and the two return series. In the interest of space, report these statistics in one table with statistics in the rows and variables in the columns. For clarity, use four decimal places for the return series.

(B) Plot the two price series over time on one single graph and describe in a few words the relationship that you observe between these two series.

(C) Some commodities tend to be in contango while other commmodities tend to be in normal backwardation.  Construct a new series correponding to the gold futures con- tract’s basis over time and plot this series on a graph. Is the gold market typically in contango or in normal backwardation? Discuss briefly.

(D) Estimate the pairwise correlation between the spot and futures returns series.  Based on this estimate, do you anticipate the gold futures contract to provide an effective hedge for your client’s gold price risk exposure? Discuss briefly.

(E) Using the risk-minimizing hedging methodology, 1) How much of her gold price risk

exposure will your client be able to eliminate thanks to the COMEX gold futures contract?  2) What proportion of her notional gold position should she hedge in the futures market?  3) Would she be better off by implementing a na¨ve hedge?  Discuss briefly.

(F) Calculate the number of futures contracts that your client should trade to minimize her gold price risk exposure. Bear in mind that one COMEX gold futures contract is for 100 troy ounces.