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.