Question 1. Duration and Banking
Consider a 5-year bond with annual coupon payments. The bond has a face value (principal) 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.
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.
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 difference between its assets and its liabilities. How does bank’s equity change, if the T-bill interest rate increases by 10 basis point?
Question 2. Option pricin
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 annualised 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%. Use discrete compounding if you need to compute semi-annual interest rate.
1. Show that there was an arbitrage opportunity with these options.
2. Construct an arbitrage strategy.
Question 3. Real option
1. What is the maximum fee the government should charge for the concession?
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. makinguse of risk-neutrality).
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 riskneutral.
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 economy. Is there any trade of atomic (Arrow-Debreu) securities possible in this economy?
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 optimal allocations and prices defined in the equilibrium).
3. At the equilibrium, calculate the forward price and risk premium for each atomic security. What do your results suggest about the consumers’ preference 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.
5. Write down the Lagrangian for the consumer’s optimisation problem, find the first order necessary conditions, and characterise the equilibrium (i.e., compute equilibrium allocations and prices of the newly available securities).
6. Now, price the newly available securities using the atomic prices from part 2. Comment on your results in light of the arbitrage-free markets.
Question 5. Investment
Suppose the risk-free rate of return is 0.03. Market porfolio 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 = r − 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