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Economics of Finance

Group Assignment

Instruction (Please read carefully.)

This assignment asks you to apply the knowledge gained in classes to analyse practical questions, present written results in a professional manner, and develop project manage- ment and teamwork skills. This group assignment should not be left until the last minute.

A suggested time-line is provided below to help to plan your work and writing. This assignment counts for 20% of the ﬁnal mark.

The deadline and submission. The deadline for submission of the group assignment is on Monday, 29 July, 10:00 am. Late submission will incur penalty upto zero mark. Please submit the assignment in the pdf format through Moodle assignment group submission in the group assignment folder. The assignment should be typed and nicely formatted. You can insert Matlab code parts to support your calculation. You may wish to insert all coding in the Appendix, or integrate in your body text. In any case, you should mind the logic 且ow and the readability of your document. Importantly, there are two other submission components relevant for this assignment: project charter and team ratings and self-evaluation.

The teamwork and Project Charter. Teamwork is an essential part of the assignment. The team project charter in the group assignment folder is a project planning and man- agement tool. Your team will be working together on planning, analysing, developing, writing up, proofreading and editing your project report ready for submission. You are also jointly responsible for allocating relevant tasks including research, logistical and or- ganisational duties. Each team member is expected to have a similar level of involvement and contribution to the assessment process and product. Each group will need to submit a charter, signed of (just typed name is su伍cient) by each team member. Each group needs to submit their ﬁlled charter, signed by each team member, in the pdf format by Monday, 15 July, 10:00 am via Moodle submission in the group assignment folder. Failure to submit Project Charter will result in group mark reduction by 5 marks. If you are not able to secure signature(s)/agreement(s) of some of the group members, it is ﬁne to submit a partially signed group charter. One person needs to submit on behalf of the group.

The team ratings and self-evaluation. After group submission, you will have a chance to rate your team member contributions and your own contribution to the group work. Each individual group member mark for group assignment will be weighted by the average of the ratings including your self-assessment. The submission will be via Moodle. In case of large discrepancies in ratings, further investigation will be conducted by the course facilitator. Submit your team ratings and self-assessment by Monday, 5 August, 10:00 am.

Plagiarism All assignments will be checked for plagiarism. See notes on Plagiarism here.

Time-line.

Week 6 Assignment released. Carefully read the assignment. Consider the method you will use, the information you will need and where to get it. A team discussion about all relevant aspects is expected. Due to social distancing, all discussion are expected to happen online, via Moodle discussion forum, Skype or https://zoom.us. To help you coordinating I suggest that the ﬁrst member of the group as posted on Moodle hosts zoom meeting in week 6 during your group tutorial time. Indeed, the group may decide on a diferent meeting organiser/time, but make sure this does not fall through the cracks. You may use https://doodle.com/ for scheduling. Break down the tasks and discuss with your team mate about potential options. Each team member is expected to do some part of each question. Draft a preliminary version of your team charter.

Week 7 Carry out a comprehensive study about relevant topics. Carefully consider each sub-question. Examine thoroughly the relevant information. Consider your presen- tation. It will be useful to draft a format template for everyone in the group to work on it. Keep a record of your progress and each individual’s contribution faithfully. Submit your team charter before the deadline.

Week 8 Solve the parts assigned to you using the knowledge from the course. Think of how your part of the question will ﬁt in the big picture. Discuss with your team mate(s) about the structure/organization of your project. Keep a record of your progress and each individual’s contribution faithfully.

Week 9 Complete your assignment. Carefully address each sub-question, and organize in a logical-coherent way. Paragraph your answer. Do not be reluctant to use full-stop. Read each sentence again. Make sure each sentence has a purpose, and it is clearly understandable. Submit your assignment before the deadline.

Submission Format. The assignment should be typed and saved as pdf, and the pdf ﬁle needs to be submitted before the deadline through the Moodle assignment submission. You may insert some parts of hand-written text/calculations if is really necessary, but make sure those are written really neatly and clearly readable. You may use https: //smallpdf.com/word-to-pdf for conversion from word to pdf if necessary.

Cover Page Each submission should contain a cover page. The cover page should specify the course “Economics of Finance”, “Group Assignment” and ﬁrst and last names of all the members of the group.

Body Text The submission should be divided into individual questions. The text should be in either 1.5 or double-spaced typescript. The font size should be no less than 12 points (use 10 point for footnotes) in a legible font. Use A4 (297mm x 210mm) paper. Include page numbers. The margins on each page should not be less than 30mm on the left-hand side, 30mm on the right-hand side, 30mm at the top and 20mm at the bottom. Diagrams, charts and tables and small code fragments should be presented within the text. If you insert a code fragment, Courier New font gives a nice look to the code.

Question 1. Duration and Banking (4 marks)

Consider a 5-year bond with annual coupon payments. The bond has a face value (prin- cipal) of $100 and sells for $95. Its coupon rate is 3%. (The coupon rate is the ratio between the coupon value and the face value). The face value is paid at the maturity year in addition to the last coupon payment.

1. Calculate the bond’s yield to maturity (YTM) and duration using its YTM. (1 mark)

2. Suppose the bond’s YTM changes in the same way as a 5-year T-bill interest rate. Use the bond’s modiﬁed duration to evaluate the relative change in the 5-year bond’s value if the interest rate on 5-year T-bills falls by one basis point, that is, by 0.0001. (1 mark)

This part was extracted from the balance sheet of the First Bank of Australia:

Assets (Billion AUD)	Liabilities (Billion AUD)
Bond 80	Fixed-rate liabilities 60

where “Bond” here refers to the bond we speciﬁed above and the ﬁxed-rate liabilities (banks future payment obligations) have an average duration of 4 years and YTM of 3%. Their YTM changes in the same way as a 5-year T-bill interest rate.

3. Bank’s equity is the diference between its assets and its liabilities. How does bank’s

equity change, if the T-bill interest rate increases by 10 basis point? (2 marks)

Question 2. Option pricing (3 marks)

BHP Billiton, the leading Australian iron ore mining giant, is listed on New York Stock Exchange. The iron ore prices have almost doubled from $67.87 on 3 August 2018 to $123.16 on 3 July 2019. The following table shows BHP stock prices in $ and the an- nualised historical volatility (Vol.) of BHP, VIX Index, the iron ore prices in $ at given dates. For simplicity, we assume one price, i.e., no bid-ask spread.

Date	BHP stock price	Vol.(%)	VIX	Iron ore price
3 Jul 2019	58.93	19.5	12.6	123.16
3 Jun 2019	52.38	18.9	18.9	93.97
3 May 2019	52.81	17.1	12.9	89.15
3 Apr 2019	56.30	19.3	13.7	87.60
4 Mar 2019	52.79	16.4	14.6	78.98
4 Feb 2019	51.12	35.3	15.7	77.85
3 Jan 2019	46.39	33.9	16.4	74.30
3 Dec 2018	46.50	36.3	20.0	67.82
5 Nov 2018	48.40	32.4	14.2	74.17
4 Oct 2018	50.01	23.2	13.2	70.79
4 Sep 2018	47.24	27.6	11.6	68.28
3 Aug 2018	50.38	31.2	16.1	67.87

A European call option on BHP stock that expired on 3 Jan 2020 and had a strike price of $65 was traded at $1.46 on 3 July 2019. A European put option on BHP stock that expired on 3 Jan 2020 and had a strike price of $65 was traded at $8.55 on 3 July 2019. The annual risk-free rate of interest was 2.15%. Using what you learnt from the course, answer the following questions. Use discrete compounding if you need to compute semi-annual interest rate.

1. Show that there was an arbitrage opportunity with these options. (1 mark)

2. Construct an arbitrage strategy. (2 marks)

Question 3. Real option (4 marks)

Option pricing methodology is often used in complex real project valuations which allow for business investment opportunities throughout the life-time of the project. Using a simple net present value (NPV) analysis for these projects may lead an incorrect valuation because NPV does not account for 且exibility of investment options.

Here is a speciﬁc example inspired by the article Sick and Gamba (2010). Suppose you are a consultant hired by a local government. The government needs to raise some cash now and hired you to determine the proper value of a three-year development concession for a speciﬁc gold-mine which has 1 million ounces of gold reserves. It is known from past practices that the gold can all be immediately produced in the year when the investment is made for a combined capital and operating cost of $290 million (this amount does not change within the three years). It is costly to store gold and it needs to be sold immediately after it is produced. Abstract from any additional ﬁnancial instruments which can be used for hedging as if they are fairly priced they will not improve our real option. A company who purchases the concession will have the right to develop the mine for the period of concession and will bear no additional tax obligations. An initial gold price is $300 per ounce. From the analysis of historical data you know that gold price will rise by 20% over a year with a probability of 0.65 and it will fall by 20% with a probability of 0.35. The riskless discount rate is 6% and the company has can develop now, or defer for either one or two years, after which point the opportunity to invest is lost (concession expires). The government asked you to produce valuation of this project from the perspective of the company (not including the concession fee since its value is not decided yet). You were asked to assume that the company is risk-neutral.

You produce the ﬁgure above to explain your reasoning. It is analogous to American option pricing. If immediately developed, the NPV of the project is $10 million, but because the concession is valid for three periods (including period 0), the government should charge a higher fee.

1. What is the maximum fee the government should charge for the concession? (1 mark)

2. Your contact person in the government, who recently studied Economic of Finance, wants you to be more explicit about your calculations. In particular you are asked to produce atomic prices for all future time-states g , b, gg , gb, bg , bb and calculate the maximum value of the project using these atomic prices and future payments. There are several ways to do this. You are free to use any method (incl. making use of risk-neutrality). (2 marks for ECON3107, 1 mark for ECON5106)

3. Discuss how the atomic prices and the project valuation would change (qualitatively, not the exact numbers), if the company was actually risk-averse rather than risk- neutral. (1 mark)

4. ECON5106 only: read the article and explain how the real options create value. (1 mark)

Reference

Sick, Gordon, and Andrea Gamba. “Some Important Issues Involving Real Options: An Overview.” Multinational Finance Journal 14, no. 1/2 (2010): 73-123.

Question 4. Arrow-Debreu Economy (7 marks)

Consider a world in which there are only two dates: 0 and 1. At date 1 there are three possible states of nature: a good weather state (G), a fair weather state (F), and a bad weather state (B). Denote S1 as the set of these states, i.e., s1 ∈ S1 = {G,F, B}. The state at date zero is known. Denote probabilities of the three states as π = (0.4, 0.3, 0.3).

There is one non-storable consumption good, apple. There are three consumers in this economy. Their preferences over apples are exactly the same and are given by the following expected utility function

where subscript. k = 1, 2, 3 denotes each consumer. In period 0, the three consumers have a linear utility and, in period 1, the three consumers have the same instantaneous utility function:

where √ = 0.2 (the coefficient of relative risk aversion). The consumers’ time discount factor, β, is 0.98.

The consumers difer in their endowments, which are given in the table below:

Endowments

t = 0 t = 1

s0 G F B Consumer 1 0.4 3.2 1.8 0.9 Consumer 2 1.2 1.6 1.2 0.4 Consumer 3 2.0 1.2 0.6 0.2

Assume that atomic (Arrow-Debreu) securities are traded in this economy. One unit of ’G security’ sells at time 0 at a price qG and pays one unit of consumption at time 1 if state ’G’ occurs and nothing otherwise. One unit of ’F security’ sells at time 0 at a price qF and pays one unit of consumption at time 1 if state ’F’ occurs and nothing otherwise. One unit of ’B security’ sells at time 0 at a price qB and pays one unit of consumption in state ’B’ only.

1. Write down the consumer’s budget constraint for all times and states, and deﬁne a Market Equilibrium in this economy. Is there any trade of atomic (Arrow-Debreu) securities possible in this economy? (1 mark)

2. Write down the Lagrangian for the consumer’s optimisation problem, ﬁnd the ﬁrst order necessary conditions, and characterise the equilibrium (i.e., compute the op- timal allocations and prices deﬁned in the equilibrium). (2 marks)

3. At the equilibrium, calculate the forward price and risk premium for each atomic security. What do your results suggest about the consumers’ preference? (1 mark)

Suppose that instead of atomic (Arrow-Debreu) securities there are three linearly independent securities, a riskless bond, a stock, and a one-period put option on this stock available for trade in this economy. The riskless bond pays 1 apple in every state, the stock pays 2, 1 and 0 apples in G, F and B, respectively. The put option has a strike price of 1.

4. Write down the budget constraint for each consumer using the newly available securities. (1 mark)

5. Write down the Lagrangian for the consumer’s optimisation problem, ﬁnd the ﬁrst order necessary conditions, and characterise the equilibrium (i.e., compute equilib- rium allocations and prices of the newly available securities). (1 mark)

6. Now, price the newly available securities using the atomic prices from part 2. Com- ment on your results in light of the arbitrage-free markets. (1 mark)

Question 5. Investment (2 marks)

Suppose the risk-free rate of return is 0.03. Market portfolio expected return is 0.10 and its risk measured by standard deviation is 0.05.

There are two investors in the economy. Their expected utility functions are given by:

Eu = e - s2 /ti , for i = 1, 2, where risk tolerance t1 = 1 and t2 = 0.5.

1. Derive the Sharpe ratio of the market portfolio. Is there a stock in the market that can beat this Sharpe ratio? (1 marks)

2. Derive the two individual investors’ portfolios. What are the expected return and risk of each individual investor’s choice? Comment on your results. (1 mark)