A market is in a calm regime or a volatile regime each day. From a calm day, the next day is volatile with probability 1/5; from a volatile day, the next day is calm with probability 1/3. The regime follows a Markov chain with these fixed probabilities. In the long run, what fraction of days are volatile? Give a fraction in lowest terms.

Learn / Library / Linear algebra

Calm and volatile days

Linear algebraEasy

A market is in a calm regime or a volatile regime each day. From a calm day, the next day is volatile with probability $\frac{1}{5}$ ; from a volatile day, the next day is calm with probability $\frac{1}{3}$ . The regime follows a Markov chain with these fixed probabilities. In the long run, what fraction of days are volatile? Give a fraction in lowest terms.

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

In the long run, the share of days moving from calm to volatile must balance the share moving back.

Solution

Let $π_{v}$ be the long-run share of volatile days, so the calm share is $π_{c} = 1 - π_{v}$ .
The stationary vector satisfies the balance $π_{c} \times \frac{1}{5} = π_{v} \times \frac{1}{3}$ .
So $\frac{1 - π _{v}}{5} = \frac{π _{v}}{3}$ , which gives $3 - 3 π_{v} = 5 π_{v}$ .
Hence $8 π_{v} = 3$ and $π_{v} = \frac{3}{8}$ .

Answer: $\frac{3}{8}$

Next question Back to the library