# Growing Dens
### Collateral Backing Ratio
The Arbera protocol uses the Collateral Backing Ratio (CBR) to measure Dens (`brTOKEN`) value relative to its underlying `TOKEN`. The CBR is calculated by dividing the `TOKEN` collateral by the total `brTOKEN` supply. As the `brTOKEN` supply decreases due to the burn fee, the CBR increases. Den tokens increase in value over time as the burn fee reduces `brTOKEN` supply, raising the `TOKEN` value per `brTOKEN` through a higher CBR.
### Scenario 1: Wrapping 2000 YEET into brYEET
**Initial Setup:**
* Initial Supply: 1000 brYEET, 1000 YEET
* Initial CBR: 1:1 (1000 YEET backing 1000 brYEET)
**User Action:**
* A new user wraps 2000 YEET into brYEET.
**Fees Applied:**
* **Wrap Fee (1.3%):** The user pays 26 brYEET as a fee.
* **Burn Fee (40% of Wrap Fee):** 10.4 brYEET is burned.
* **Remaining after Burn:** 15.6 brYEET is redistributed.
**Outcome:**
* **Total brYEET Supply:** 3000 brYEET (1000 initial + 2000 new - 0 reduction due to fee redistribution)
* **After Burn:** 2989.6 brYEET (3000 brYEET - 10.4 brYEET burned)
* **Total YEET Backing:** 3000 YEET (1000 initial + 2000 new)
**New Collateral Backing Ratio (CBR):**
* **New CBR:**\
$$\frac{3000 \text{ YEET}}{2989.6 \text{ brYEET}} = 1.003478:1$$
**Explanation:**
* The CBR increases slightly because the burn fee reduces the total brYEET supply, while the total YEET backing remains at 3000, resulting in each brYEET being backed by slightly more than 1 YEET.
### Scenario 2: Unwrapping 1000 brYEET into YEET
**Initial Setup:**
* Initial Supply: 2989.6 brYEET, 3000 YEET
* Initial CBR: 1.003478:1 (3000 YEET backing 2989.6 brYEET)
**User Action:**
* A user unwraps 1000 brYEET into YEET.
**Fees Applied:**
* **Unwrap Fee (0.7%):** The user pays 7 brYEET as a fee.
* **Burn Fee (40% of Unwrap Fee):** 2.8 brYEET is burned.
* **Remaining after Burn:** 4.2 brYEET is redistributed.
**Outcome:**
* **Total brYEET Supply:** 1989.6 brYEET (2989.6 initial - 1000 unwrapped)
* **After Burn:** 1986.8 brYEET (1989.6 brYEET - 2.8 brYEET burned)
* **Total YEET Backing:** 1996.52 YEET (3000 initial - 1003.48 YEET unwrapped)
**New Collateral Backing Ratio (CBR):**
* **New CBR:**\
$$\frac{1996.52 \text{ YEET}}{1986.8 \text{ brYEET}} = 1.00489:1$$
**Explanation:**
* The CBR increases slightly because the burn fee during unwrapping reduces the brYEET supply more significantly than the reduction in YEET backing, resulting in each remaining brYEET being backed by more YEET.
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