Answer: 0.16

## Explanation The Power Law states that for many financial variables, the probability that the variable exceeds a certain value x is given by: $$P(v > x) = Kx^{-\alpha}$$ Where: - $\alpha$ is the power law exponent - $K$ is a constant **Step 1: Find the constant K** We are given: - $\alpha = 4$ - $P(v > 20) = 0.01$ Using the Power Law formula: $$0.01 = K \times 20^{-4}$$ $$0.01 = K \times \frac{1}{20^4}$$ $$0.01 = K \times \frac{1}{160,000}$$ $$K = 0.01 \times 160,000 = 1,600$$ **Step 2: Calculate $P(v > 10)$** Now using the same formula with $x = 10$: $$P(v > 10) = K \times 10^{-4}$$ $$P(v > 10) = 1,600 \times \frac{1}{10^4}$$ $$P(v > 10) = 1,600 \times \frac{1}{10,000}$$ $$P(v > 10) = 1,600 \times 0.0001$$ $$P(v > 10) = 0.16$$ **Step 3: Interpretation** The probability that the financial variable exceeds 10 is 0.16 or 16%. This makes sense because: - When $x$ increases from 10 to 20 (doubling), the probability decreases by a factor of $2^4 = 16$ - $0.16 / 0.01 = 16$, which confirms the power law relationship **Key Insight**: The Power Law shows that extreme events (large values of v) become increasingly rare according to a specific mathematical pattern. In this case, with $\alpha = 4$, the probability decays rapidly as the threshold increases.

Author: Nikitesh Somanthe