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A hedge fund classifies each trading day into one of three market regimes:
\[
B = \text{bullish}, \qquad
N = \text{neutral}, \qquad
R = \text{bearish}.
\]
The regime evolves as a Markov chain with transition matrix
\[
P =
\begin{pmatrix}
0.60 & 0.30 & 0.10 \\
0.25 & 0.50 & 0.25 \\
0.15 & 0.35 & 0.50
\end{pmatrix}
\]
where rows represent the current state and columns represent the next state in the order
\[
(B, N, R).
\]
At the start of day \(1\), the market is neutral.
Find the probability that the market is bearish on day \(4\).
Check your answer
Solution
Answer: 27%
Let the state probability vector for day \(n\) be
\[
\pi_n = (p_B, p_N, p_R).
\]
Since the market starts neutral,
\[
\pi_1 = (0,1,0).
\]
The Markov evolution is
\[
\pi_{n+1} = \pi_n P.
\]
Compute the distribution for each day.
Day \(2\):
\[
\pi_2 = (0,1,0)P
\]
\[
= (0.25,\,0.50,\,0.25).
\]
Day \(3\):
\[
\pi_3 = \pi_2 P
\]
Compute each component.
Bullish:
\[
0.25(0.60) + 0.50(0.25) + 0.25(0.15) = 0.3125.
\]
Neutral:
\[
0.25(0.30) + 0.50(0.50) + 0.25(0.35) = 0.4125.
\]
Bearish:
\[
0.25(0.10) + 0.50(0.25) + 0.25(0.50) = 0.275.
\]
So
\[
\pi_3 = (0.3125,\,0.4125,\,0.275).
\]
Now compute day \(4\):
\[
\pi_4 = \pi_3 P.
\]
We only need the bearish component.
\[
p_R(4)
=
0.3125(0.10)
+
0.4125(0.25)
+
0.275(0.50).
\]
\[
p_R(4) = 0.271875.
\]
Thus the probability that the market is bearish on day \(4\) is 0.27.