Goldman Sachs Commodity Analyst John Roberts Wanted to Use Markov $>=0.50 ; \mathrm{N}$

Essay

Goldman Sachs commodity analyst John Roberts wanted to use Markov Chains to analyze the price movement of gold. He looked at the changes in the closing price per ounce of gold bullion in the Chicago Board of Trade for any two consecutive trading days. He recognized three states: U (up): today's closing price - previous closing price $>=0.50 ; \mathrm{N}$ (neutral): $-\$ 0.50<$ today's closing price - previous closing price $<\$ 0.50$ ; and D (down): today's closing price - previous closing price $<=-\$ 0.50$ . John goes on to construct a transition matrix based on these state definitions; the matrix is given below.

(A) Draw a tree diagram showing the choices for two periods, starting from $\mathrm{U}$ and $\mathrm{N}$ .
(B) If the current state of the bullion market were $U$ , what would be the probability distribution for the states occupied by the bullion market after 2 trading days?
(C) What would be the long-run proportions (steady-state probabilities) for each state?
(D) In a five-year period with 1,201 trading days, how many of these days would you expect the system to be in U state?