Simulation Mechanics

This tool visualizes how the market's correlation structure affect CAPM βs in the cross-section.

We simulate stock $i$'s CAPM $\beta_i$ as

$$ \beta_i = \frac{\sigma_i}{\sigma_M^2} \sum_{j=1}^{N} w_j \sigma_j \color{red}{\rho_{ij}} $$

where:

• $w_j$: Market weight of stock $j$
• $\color{red}{\rho_{ij}}$: Pairwise correlation between $i$ and $j$
• $\sigma_M^2$: Total market variance
• $\sigma_i$: Volatility of stock $i$
  ($\sigma_i=0.3 \ \forall \ i \in N$)

Simulation Ingredients

1. Market Structure (Log-Normal):
Stock markets have a heavy-tailed size distribution. We simulate this using a Log-Normal($\mu, \sigma$) distribution for market caps. The U.S. stock market can be approximated with $N=3500$ stocks and $\sigma\approx2$, such that the 1,000 largest firms account for about 90% of total market capitalization.

2. Correlation Matrix ($\rho$):
We assume a baseline "global" correlation for all stock pairs.

3. Comovement Shocks (+$\Delta\rho$):
Passive flows cause stocks within an index to move together, increasing their pairwise correlations $\rho_{ij}$. Use the Shock sliders to apply this excess comovement to specific rank ranges (e.g., stocks ranked 1,000–3,000).