# Question: Each year Ms Fontanez has the chance to invest in

Each year Ms. Fontanez has the chance to invest in two different no-load mutual funds: the Go-Go Fund or the Go-Slow Mutual Fund. At the end of each year, Ms. Fontanez liquidates her holdings, takes her profits, and then reinvests. The yearly profits of the mutual funds depend on where the market stood at the end of the preceding year. Recently the market has been oscillating around the 12,000 mark from one year end to the next, according to the probabilities given in the following transition matrix:

Each year that the market moves up (down) 1,000 points, the Go- Go Fund has profits (losses) of $20,000, while the Go-Slow Fund has profits (losses) of $10,000. If the market moves up (down) 2,000 points in a year, the Go-Go Fund has profits (losses) of $50,000, while the Go-Slow Fund has profits (losses) of only $20,000. If the market does not change, there is no profit or loss for either fund. Ms. Fontanez wishes to determine her optimal investment policy in order to minimize her (long-run) expected average cost (loss minus profit) per year.

(a) Formulate this problem as a Markov decision process by identifying the states and decisions and then finding the Cik.

(b) Identify all the (stationary deterministic) policies. For each one, find the transition matrix and write an expression for the (long-run) expected average cost per period in terms of the unknown steady-state probabilities (π0, π1, . . . , πM).

(c) Use your IOR Tutorial to find these steady-state probabilities for each policy. Then evaluate the expression obtained in part (b) to find the optimal policy by exhaustive enumeration.

Each year that the market moves up (down) 1,000 points, the Go- Go Fund has profits (losses) of $20,000, while the Go-Slow Fund has profits (losses) of $10,000. If the market moves up (down) 2,000 points in a year, the Go-Go Fund has profits (losses) of $50,000, while the Go-Slow Fund has profits (losses) of only $20,000. If the market does not change, there is no profit or loss for either fund. Ms. Fontanez wishes to determine her optimal investment policy in order to minimize her (long-run) expected average cost (loss minus profit) per year.

(a) Formulate this problem as a Markov decision process by identifying the states and decisions and then finding the Cik.

(b) Identify all the (stationary deterministic) policies. For each one, find the transition matrix and write an expression for the (long-run) expected average cost per period in terms of the unknown steady-state probabilities (π0, π1, . . . , πM).

(c) Use your IOR Tutorial to find these steady-state probabilities for each policy. Then evaluate the expression obtained in part (b) to find the optimal policy by exhaustive enumeration.

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