Consider an economy in three periods, t = 0, t = 1 and t = 2. At t = 0, the market index is trading at a value of 100. At t=1, the index either rises by 30 or falls by 10 with equal probabilities. Following an increase at t=1, the index either increases by 30 with probability 1/4, or falls by 10 with probability 3/4 at t=2. Aftera fall at t=1, the index either increases by 30 with probability 3/4, or falls by 10 with probability 1/4, by t = 2. The index pays no dividends, and the riskfree rate in eachperiod is rf= 0.
(a) Draw the event tree of this economy. For both nodes at t=1, compute the netindex return between periods 0 and 1. What is the expected return of the indexbetween t = 0 and t = 1?
(b) For each node at t = 2, compute the probability of reaching that node and therealized index return between t = 0 and t = 2. What is the expected return ofthe index between t = 0 and t = 2? What is the mean and variance of the indexreturn between t = 0 and t = 2?
(c) Suppose that you wish to form a portfolio of the market index and the riskfreeasset at t=0 and hold it until t = 2 (no rebalancing at t = 1). If you are amean-variance optimizer with risk aversion A = 5, how should you invest?
(d) Now consider the following \market-timing" investment strategy. At t = 0, youinvest $100 in the market index. At t = 1, if the market has gone up, sell all ofyour shares in the market index and invest everything in the riskfree asset. If themarket has gone down at t=1, then continue to hold $100 in the market index.
What is the expected return between t = 0 and t = 2 of the strategy? Is it higheror lower than the expected return in (b)?
(e) What is the variance of the return from the strategy in part (d)? Is it higher orlower than the variance of the strategy in (b)? Which of these two strategies wouldyour mean-variance optimizing investor choose? Does market timing - adjustingthe portfolio when expected returns change - benefit the investor?