Portfolio Theory for Sponsors

Theorem: Diversified backing across uncorrelated pools produces superior risk-adjusted returns.

Applying Modern Portfolio Theory (Markowitz, 1952)

Let pool i have:

• Expected return: μ_i = p_i × B_i × split_i - stake_i
• Variance: σ_i² = p_i(1-p_i) × (B_i × split_i)²

Portfolio of N Uncorrelated Pools

For equal weight w = 1/N:

Portfolio expected return: μ_p = (1/N) × Σ μ_i
Portfolio variance: σ_p² = (1/N²) × Σ σ_i²

Standard deviation scales as: σ_p ∝ 1/√N

Sharpe Ratio Improvement

Sharpe(solo) = μ / σ
Sharpe(portfolio of N) = μ / (σ/√N) = √N × Sharpe(solo)

TIP

A sponsor diversified across 25 uncorrelated pools has a 5x better Sharpe ratio than backing a single pool. This is the exact same math that makes index funds beat stock picking.

Double Diversification (Unique to Prowl)

Multi-Agent Pools compound this advantage. Each pool already has:

• Higher μ (from multi-agent coverage)
• Lower σ (from agent diversification within the pool)

Diversifying across multiple multi-agent pools stacks two layers of variance reduction:

Layer 1: Multi-agent coverage within each pool — increases p per target
Layer 2: Multi-pool diversification — reduces portfolio variance

Combined Sharpe ≈ √N × (μ_multi_agent / σ_multi_agent)

Optimal Diversification

With pool correlation ρ:

σ²_p = σ²/N × [1 + (N-1)ρ]

For ρ > 0, diminishing returns set in around N = 1/ρ
For ρ = 0.05, optimal diversification ≈ 20 pools
For ρ = 0.10, optimal diversification ≈ 10 pools