A regime V takes 3 values with probabilities 1/2, 1/4 and 1/4. The size of the daily return is 1, 2 or 4 in those regimes, and its sign is plus or minus with chance 1/2 each, whatever the regime. So the return R has zero correlation with the size. What is I(R; V) in bits? Give a fraction.

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Regimes that correlation misses

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A regime $V$ takes $3$ values with probabilities $\frac{1}{2}$ , $\frac{1}{4}$ and $\frac{1}{4}$ . The size of the daily return is $1$ , $2$ or $4$ in those regimes, and its sign is plus or minus with chance $\frac{1}{2}$ each, whatever the regime. So the return $R$ has zero correlation with the size. What is $I (R; V)$ in bits? Give a fraction.

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Hint

Does $R$ leave any doubt about $V$ ?

Solution

The sizes differ, so $R$ fixes the regime and $H (V ∣ R) = 0$ .
So $I (R; V)$ is $H (V)$ , which is $\frac{1}{2} \times 1 + \frac{1}{4} \times 2 + \frac{1}{4} \times 2 = \frac{3}{2}$ .

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

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