# Agent Pear Statistics

Users of the Pear Protocol front-end will see a collapsible "Agent Pear" section that houses some advanced statistics. 

{% hint style="info" %}
You do not need to understand these terms to be a profitable pair trader. Most users will not need to reference these values, but we include them for users who may find them a helpful reference point.
{% endhint %}

Here is a wider definition of each of these terms.

## 1. Correlation

Correlation measures the strength and direction of the relationship between the hourly price movements of the two assets. We use **Pearson’s correlation coefficient**, which ranges from **-1** to **+1**.

* **+1** → perfect positive correlation (assets move together).
* **-1** → perfect negative correlation (assets move in opposite directions).
* **0** → no linear relationship.

Correlation helps you analyse if the pairs selected have a directional relationship and the magnitude of that relationship.

## 2. Cointegration

While correlation captures short-term co-movements, **cointegration** tests for a *long-term, mean-reverting relationship* between two asset prices.

* Assets may be highly correlated but still drift apart (non-stationary spread).
* Cointegration ensures that, despite short-term deviations, the spread between the assets tends to revert to a stable mean.

This property underpins the profitability of statistical arbitrage: if the spread diverges, it has a statistical tendency to revert. Again, cointegration is not a necessity for most pair traders unless you are doing statistical arbitrage.

## 3. Rolling Z-score

The **rolling Z-score** measures how far the current spread is from its rolling mean, in units of standard deviations:

$$
Z = \frac{\text{Spread} - \text{Mean(Spread)}}{\text{StdDev(Spread)}}
$$

* **High positive Z-score** → spread is wider than usual (potentially short this pair).
* **Low negative Z-score** → spread is tighter than usual (potentially long this pair).
* **Z-score near 0** → spread is at equilibrium.

Traders typically use thresholds (e.g., enter at ±2, exit at 0) to systematize when to enter and exit mean reverting pair trades.

## 4. Beta

In trading, **beta (β)** measures how much one asset moves relative to another. In our case, beta represents the **rolling hedge ratio**, which determines how to size the long and short legs of a trade to maintain a **beta-neutral position**.

* Beta measures how much one asset moves relative to the other.
* Example: if Asset A has a **beta of 1.26** versus Asset B, this means that for every 1% move in B, Asset A tends to move 1.26%.

**Practical example:**\
\
Constructing a **long A / short B** pair with $100,000 capital:

**Practical example:**

Constructing a **long A / short B** pair with $100,000 capital:

$$
1.26 \times (\text{Long A}) = \text{Short B}
$$

$$
\text{Long A} \approx $44,248, \quad \text{Short B} \approx $55,752
$$

This weighting ensures the portfolio is **beta-neutral**, so P\&L is driven primarily by the convergence of the spread rather than market direction. In an ideal scenario a trader is rebalancing their trades as the beta evolves over-time.

## 5. Volatility 

Volatility measures the **standard deviation of log returns** for the pair. It reflects the typical magnitude of price swings in the spread.

σ = √( 1 ÷ (N–1) × Σ ( rᵢ – r̄ )² )

where rᵢ are the log returns of the spread and r̄ is their mean.

* Higher volatility → larger and more frequent price swings (more opportunity, more risk)
* Lower volatility → smaller price fluctuations (steadier spreads, fewer signals)

Volatility helps traders calibrate position sizing and risk management for each pair.

⚡ **Together:**

* **Correlation** → validates short-term co-movement.
* **Cointegration** → confirms long-term mean reversion.
* **Rolling Z-score** → provides entry/exit signals.
* **Beta** → ensures correct hedge sizing and risk balance.
* **Volatility** → quantifies the riskiness of the spread and informs overall position sizing