Consider an economy over three periods. At t = 0, the stock market index is priced at
Question:
Consider an economy over three periods. At t = 0, the stock market index is priced at 100. At t = 1, the index either rises by 30 or falls by 10 with equal probabilities. Following a rise at t = 1, the index either rises by 30 with probability 0.25, or falls by 10 with probability 0.75 at t = 2. Following a fall at t = 1, the index either rises by 30 with probability 0.75, or falls by 10 with probability 0.25. Thus, the highest possible index value at t = 2 is 160, and the lowest possible index value at t = 2 is 80. The stock index pays no dividends, and the risk-free rate in each period is Rf = 0.
- Draw the event tree for the stock index in this economy. Include the price of the stock index for each node and the probability for each branch.
- For both nodes at t = 1, compute the net simple return between periods 0 and 1. What is the expected return of the index between t = 0 and t = 1?
- For each node at t = 2, compute the probability of reaching that node, and the net simple return between periods 0 and 2. What is the expected return and the variance of the return between t = 0 and t = 2?
- Suppose you form a portfolio of the risky stock index and the risk-free asset at t = 0 and hold it until t = 2 with no rebalancing. If you are a mean-variance optimiser with risk aversion coefficient A = 5, how should you invest? [Hint: Recall that a mean-variance optimiser seeks to maximise .]
- Now, suppose you follow a “market timing” strategy. At t = 0, invest 100% in the stock index. At t = 1, if the market has gone up, sell all your holdings and invest everything in the risk-free asset. If the market is down at t = 1, then continue to hold 100% in the stock index. If you started with £100 at t = 0, what is the net simple return between periods 0 and 2 for each possible outcome at t = 2?
f. What is the expected return and the variance of the return for the strategy in part (e)? As a risk-averse mean-variance optimising investor, would you prefer the buy and hold strategy of part (d) or the market timing strategy of part (e)?
Introduction to Derivatives and Risk Management
ISBN: 978-1305104969
10th edition
Authors: Don M. Chance