Econ Question

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. After a 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 each period is r f = 0.

(a) Draw the event tree of this economy. For both nodes at t=1, compute the net index return between periods 0 and 1. What is the expected return of the index between t = 0 and t = 1?

(b) For each node at t = 2, compute the probability of reaching that node and the realized index return between t = 0 and t = 2. What is the expected return of the index between t = 0 and t = 2? What is the mean and variance of the index return between t = 0 and t = 2?

(c) Suppose that you wish to form a portfolio of the market index and the riskfree asset at t=0 and hold it until t = 2 (no rebalancing at t = 1). If you are a mean-variance optimizer with risk aversion A = 5, how should you invest?

(d) Now consider the following \market-timing" investment strategy. At t = 0, you invest $100 in the market index. At t = 1, if the market has gone up, sell all of your shares in the market index and invest everything in the riskfree asset. If the market 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 higher or lower than the expected return in (b)?

(e) What is the variance of the return from the strategy in part (d)? Is it higher or lower than the variance of the strategy in (b)? Which of these two strategies would your mean-variance optimizing investor choose? Does market timing – adjusting the portfolio when expected returns change - benefit the investor?