The price of a certain stock is fluctuating between $10,

$20, and $30 from month to month. Market analysts have predicted that if the stock is at $10 during any month, it will be at $10 or

$20 the next month, with probabilities 4

5 and 1

5

, respectively; if the stock is at $20, it will be at $10, $20, or $30 the next month, with probabilities 1

4

, 1 4

, and 1

2

, respectively; and if the stock is at $30, it will be at $20 or $30 the next month, with probabilities 3

4 and 1

4

, respectively. Given a discount factor of 0.9, use the policy improve ment algorithm to determine when to sell and when to hold the stock to maximize the expected total discounted profit. (Hint: Include a state that is reached with probability 1 when the stock is sold and with probability 0 when the stock is held.)