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Long-run sentiment share

ProbabilityEasy

A stock's daily sentiment is either Up or Down. If today is Up, tomorrow is Up with probability $0.8$ . If today is Down, tomorrow is Up with probability $0.4$ . Find the long-run fraction of Up days, as a fraction a/b in lowest terms.

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Hint

In the long run, the flow from Up to Down must equal the flow from Down to Up.

Solution

Let $π$ be the long-run fraction of Up days, so Down days have fraction $1 - π$ .
The flow from Up to Down is $0.2 π$ per day, and the flow from Down to Up is $0.4 (1 - π)$ .
In equilibrium these flows are equal: $0.2 π = 0.4 (1 - π)$ .
This gives $0.6 π = 0.4$ , so the fraction of Up days is $\frac{2}{3}$ .

Answer: $\frac{2}{3}$

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