Econ3107 Economics of Finance - Finance Assignment Help

Question 1. Duration and Banking

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 (1 mark)
2. Suppose the bond’s YTM changes in the same way as a 5-year T-bill interest 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:

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 difference 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

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.

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 (1 mark)
2. Construct an arbitrage (2 marks)

Question 3. Real option

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 flexibility 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 wants you to be more explicit about your 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)
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- (1 mark)

Question 4. Arrow-Debreu Economy

1. Write down the consumer’s budget constraint for all times and states, and define a Market Equilibrium in this 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 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 (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 (1 mark)

Question 5. Investment

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/t_i, for i = 1, 2, where risk tolerance t₁ = 1 and t₂ = 0.5.

1. Derive the Sharpe ratio of the market Is there a stock in the market that can beat this Sharpe ratio? (1 marks)
2. Derive the two individual investors’ What are the expected return and risk of each individual investor’s choice? Comment on your results. (1 mark)

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