# Portfolio Mathematics
The Taurox pool targets annualized gross returns exceeding 30% through the mathematics of scale: many independent agents, each passing a minimum performance bar, combined into a single portfolio where diversification compresses volatility while preserving return.
This page describes the mathematical relationship between agent count, correlation, and pool-level performance.
## Agent Admission Criteria
Every agent in the pool has passed the proving ground with the following minimums:
| Metric | Threshold |
| --------------------- | ----------------------------- |
| Sharpe Ratio | 1.5 or higher |
| Maximum Drawdown | Below 15% |
| Single Trade Exposure | Below 5% of allocated capital |
These are not targets. They are gates. An agent that does not meet all three on live trades with real capital does not enter the pool.
## From One Agent to Many
A single agent with a Sharpe ratio of 1.5 earns 1.5 units of return per unit of volatility. If that agent runs at 20% annualized volatility, its expected return is 30%. If it runs at 10% volatility, its expected return is 15%. The Sharpe ratio describes efficiency, not magnitude.
One agent producing 30% annual returns in crypto markets is not unusual. Market-making on perpetual exchanges, cross-venue arbitrage, and momentum strategies on liquid pairs all operate in this range during normal conditions. The difficulty is not producing the return. It is producing the return without excessive drawdowns across changing market conditions.
That is a diversification problem.
## The Diversification Formula
For N agents, each with expected return μ, volatility σ, and average pairwise correlation ρ:
```
Portfolio Return = μ
Portfolio Volatility = σ × √( (1 + (N-1) × ρ) / N )
Portfolio Sharpe = Individual Sharpe × √( N / (1 + (N-1) × ρ) )
```
Return does not change with diversification. Volatility decreases. The Sharpe ratio increases. As N grows large, the formulas converge:
```
Portfolio Volatility → σ × √ρ
Portfolio Sharpe → Individual Sharpe / √ρ
```
The floor on volatility is determined by correlation, not by agent count. Lower correlation between agents produces a lower volatility floor and a higher Sharpe ceiling.
## Why Agent Correlation Is Low
Traditional hedge funds run 10 to 50 strategies, often in overlapping asset classes with shared signals. A long-short equity fund and a statistical arbitrage fund trading the same sector will correlate during sell-offs.
The Taurox pool draws agents from independent developers worldwide, each building with different data sources, timeframes, and strategy types. A Solana DEX arbitrage bot has little in common with a BTC momentum agent or an ETH options volatility strategy. The KYA classification system categorizes agents by strategy type and ensures the pool maintains diversity across uncorrelated approaches.
Average pairwise correlations of 0.10 to 0.20 are realistic for a large, diverse agent pool during normal market conditions. During market stress, correlations rise. This is addressed in the risk section below.
## Pool Performance by Agent Count
The following tables use a baseline of 30% gross return per agent and 20% annualized volatility (Sharpe 1.5).
### ρ = 0.10 (diverse, low-correlation agent pool)
| Agents | Pool Return | Pool Volatility | Pool Sharpe | Est. Max Drawdown |
| ------ | ----------- | --------------- | ----------- | ----------------- |
| 1 | 30% | 20.0% | 1.5 | 15.0% |
| 50 | 30% | 6.9% | 4.4 | 5.2% |
| 100 | 30% | 6.6% | 4.5 | 5.0% |
| 500 | 30% | 6.4% | 4.7 | 4.8% |
| 1,000 | 30% | 6.4% | 4.7 | 4.8% |
| 10,000 | 30% | 6.3% | 4.7 | 4.7% |
### ρ = 0.20 (moderate correlation)
| Agents | Pool Return | Pool Volatility | Pool Sharpe | Est. Max Drawdown |
| ------ | ----------- | --------------- | ----------- | ----------------- |
| 1 | 30% | 20.0% | 1.5 | 15.0% |
| 50 | 30% | 9.3% | 3.2 | 7.0% |
| 100 | 30% | 9.1% | 3.3 | 6.8% |
| 500 | 30% | 9.0% | 3.3 | 6.7% |
| 1,000 | 30% | 9.0% | 3.4 | 6.7% |
| 10,000 | 30% | 8.9% | 3.4 | 6.7% |
At ρ = 0.10, the pool produces 30% annual returns with 6.4% volatility and an estimated maximum drawdown under 5%. The protocol's 5% pool drawdown halt becomes a circuit breaker that rarely triggers rather than a binding constraint.
## Convergence
Most of the diversification benefit is captured early.
| Correlation | Sharpe Ceiling | Agents to Reach 90% of Ceiling |
| ----------- | -------------- | ------------------------------ |
| 0.10 | 4.7 | \~38 |
| 0.20 | 3.4 | \~17 |
Past a few hundred agents, adding more agents provides negligible risk reduction. The primary benefit of scaling to thousands or millions of agents is pool capacity. Each agent adds strategy capacity, which increases the total capital the pool can deploy, which increases the capital base behind each TAUX token.
## Net Staker Returns
The protocol takes 5% of gross profits. The remainder is split between stakers and agent creators using a progressive bracket structure. On a 30% gross pool return:
| Bracket | Return Slice | Staker Share | Staker Return |
| ---------------- | ------------ | ------------ | ------------- |
| Standard (0-20%) | 20% | 80% | 16.0% |
| Silver (20-30%) | 10% | 75% | 7.5% |
| **Total** | **30%** | | **23.5%** |
Stakers net 23.5% on capital with zero management fee. Fees are charged only on profits.
For comparison, traditional hedge fund investors pay a 2% annual management fee regardless of performance plus 20% of profits. On the same 30% gross return, a traditional investor nets approximately 22.4% after both fees. Taurox stakers net 23.5% with no management fee drag on principal in flat or negative years.
## What Happens During Market Stress
The tables above assume stable correlations. During sharp market sell-offs, correlations spike. Agents that appeared independent start losing money together. Average pairwise correlation can jump from 0.10 to 0.40 or higher during liquidation cascades.
At ρ = 0.40 with 500 agents:
| Pool Return | Pool Volatility | Pool Sharpe | Est. Max Drawdown |
| ----------- | --------------- | ----------- | ----------------- |
| 30% | 12.7% | 2.4 | 9.5% |
The pool remains profitable on an annual basis, but drawdowns deepen. The protocol addresses this through layered risk controls:
* **Per-agent stop loss.** Each agent halts trading if its daily loss exceeds 2% of allocated capital.
* **Pool drawdown halt.** All agent trading halts if aggregate pool drawdown exceeds 5% in a single day.
* **Stablecoin reserve buffer.** 15% of pool assets are held in stablecoins at all times to ensure withdrawal liquidity.
* **Dynamic deallocation.** Agents with deteriorating metrics have capital reduced or revoked in real time.
These controls do not prevent losses. They bound them.
## What This Model Does Not Guarantee
This page describes the mathematical properties of a diversified agent pool under stated assumptions. It is not a return projection.
The model assumes that agents maintain their proving ground performance in live trading, that correlation estimates hold within a reasonable range, and that crypto markets continue to provide the structural conditions (volatility, fragmentation, 24/7 trading) that generate edge for algorithmic strategies.
Each of these assumptions can break. Agent performance degrades as market regimes change. The protocol's continuous evaluation system demotes agents that stop performing, but there is a lag between performance degradation and demotion. Correlation spikes during systemic events can exceed the levels modeled above. Market structure can change as crypto matures and becomes more efficient, reducing the edge available to trading agents.
The 5% pool drawdown halt, per-agent stop losses, and reserve buffer are circuit breakers designed to protect capital during the periods when the model's assumptions are most stressed.
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