CALCULATING APR

1. Understanding APR in Pyramid Financial

In the Pyramid Financial ecosystem, the Annual Percentage Rate (APR) represents the yearly rate of return that participants earn by staking their PYRA tokens. The APR is designed to remain highly attractive (approximately 1000%) while incorporating a dynamic adjustment mechanism to ensure sustainability as more tokens are staked.

2. APR Calculation Formula

To maintain an APR close to 1000% even as a significant portion of the PYRA token supply is staked, we implement a modified linear adjustment formula. This formula ensures that the APR decreases slightly in response to increased staking but remains predominantly around 1000%.

APR Calculation:

APR = 1000% - (Total Staked / Total Supply) * 10%

Where:

• 1000% is the base APR when no tokens are staked.

• Total Staked is the number of PYRA tokens currently staked.

• Total Supply is the current total number of PYRA tokens in circulation (after burning staked tokens).

• 10% is the adjustment factor that controls how much the APR decreases relative to the staking ratio.

Key Characteristics:

• Inverse Relationship: As the Total Staked increases relative to the Total Supply, the APR decreases linearly.

• Controlled Reduction: The APR decreases by 1% for every 10% increase in the staking ratio, ensuring that it remains around 1000% even with substantial staking activity.

• Sustainability: This mechanism balances rewarding participants generously while preventing the overinflation of rewards as staking grows.

3. Mechanics Behind the Formula

• Initial APR: When no tokens are staked, the APR is at its maximum (1000%), incentivizing early participation.

• Dynamic Adjustment: As more tokens are staked, the APR decreases proportionally, maintaining a high incentive without compromising the platform's financial health.

• Reward Distribution: Rewards are minted in PYRA tokens based on the calculated APR and the staked amount, aligning incentives with platform growth.

4. Example Calculation: Staking 10 PYRA Tokens

To illustrate how APR is calculated and how rewards are distributed, let's consider a participant who stakes 10 PYRA tokens.

Assumptions:

• Initial Total Supply: 10,000 PYRA tokens

• ETH Pool Value: 1 ETH = $2,500

• PYRA Token Price: 1 PYRA = $250

• Participant's Stake: 10 PYRA tokens

• Epoch Duration: 6 hours

• Adjustment Factor: 10% (as per the APR formula)

Step-by-Step Process:

Step 1: Initial State Before Staking

• Total Supply: 1,000 PYRA

• Total Staked: 0 PYRA

• APR Calculation:

APR = 1000% - (0 / 10000) * 10%

• Annual Rewards for Staking 10 PYRA:

Annual Rewards = 10 PYRA * 1000%

Annual Rewards = 100 PYRA tokens

Step 2: Participant Stakes 10 PYRA Tokens

• Action: The participant stakes (burns) 10 PYRA tokens.

• Total Supply After Staking:

New Total Supply = Initial Total Supply - Staked Amount

New Total Supply = 10000 PYRA - 10 PYRA

New Total Supply = 9990 PYRA

Total Staked = 10 PYRA

Step 3: APR Recalculation After Staking

APR = 1000% - (10 / 9990) * 10%

APR ≈ 999.90%

• New APR: Approximately 999.90%

Step 4: Reward Calculation Per Epoch

To determine the rewards per epoch, we prorate the APR based on the epoch duration.

• Annual APR: 999.90%

• Epoch Duration: 6 hours

• Total Epochs in a Year: 1460

• Epoch APR:

Epoch APR = Annual APR / Total Epochs

Epoch APR = 999.90% / 1460 Epoch

APR ≈ 0.685% per epoch

• Rewards for Staking 10 PYRA Per Epoch:

Epoch Rewards = 10 PYRA * 0.685%

Epoch Rewards ≈ 0.0685 PYRA

• USD Value of Epoch Rewards (if 1 PYRA = $250):

USD Rewards = 0.0685 PYRA * $250/PYRA

USD Rewards ≈ $17.12

Step 5: Reward Distribution Mechanism

• Reward Tokens: The platform mints 0.0685 PYRA tokens as rewards for the participant at the end of each epoch.

• Tokenomics Impact:

  • Total Supply After Rewards:

Total Supply = 9990 PYRA (after staking) + 0.0685 PYRA (rewards)

Total Supply = 9990.0685 PYRA

• New Total Staked: Remains 10 PYRA

Summary of the Example

Action	PYRA Tokens	Calculation	Result
Initial APR	-	APR = 1000% - (0 / 10000) * 10% = 1000%	1000%
Staking 10 PYRA	10	Burns 10 tokens	Total Supply: 9990 PYRA
APR After Staking	-	APR = 1000% - (10 / 9990) * 10% ≈ 999.90%	~999.90%
Epoch APR	-	Epoch APR = 999.90% / 1460 ≈ 0.685% per epoch	~0.685% per epoch
Annual Rewards	-	Annual Rewards = 10 PYRA * 1000% = 100 PYRA	100 PYRA annually
Epoch Rewards	-	Epoch Rewards = 10 PYRA * 0.685% ≈ 0.0685 PYRA	0.0685 PYRA per epoch
USD Value of Epoch Rewards	-	USD Rewards = 0.0685 PYRA * $250 ≈ $17.12	$17.12 per epoch
Total Supply After Rewards	-	9990 PYRA + 0.0685 PYRA = 9990.0685 PYRA	9990.0685 PYRA

5. Visual Representation

To further clarify the APR dynamics and reward distribution, consider the following table showcasing different staking scenarios:

Total Staked (PYRA)	Total Supply (PYRA)	APR (%)
0	10,000	1000%
50	9,950	~999.50%
500	9,500	1000% - (500 / 9500) * 10% ≈ 994.74%
1000	9,000	1000% - (1000 / 9000) * 10% ≈ 988.89%
2000	8,000	1000% - (2000 / 8000) * 10% = 975%
3000	7,000	1000% - (3000 / 7000) * 10% ≈ 957.14%
4000	6,000	1000% - (4000 / 6000) * 10% ≈ 933.33%
4500	5,500	1000% - (4500 / 5500) * 10% ≈ 918.18%
4900	5,100	1000% - (4900 / 5100) * 10% ≈ 903.92%
5000	5,000	1000% - (5000 / 5000) * 10% = 900%

6. Key Takeaways

1. High Initial APR: When fewer tokens are staked, the APR remains at its maximum (1000%), providing strong incentives for early participants to stake their tokens.

2. Minimal APR Adjustment: As tokens are staked, the APR decreases slightly (by 0.101% for staking 10 PYRA out of 990 PYRA), ensuring it remains consistently high (~1000%).

3. Reward Distribution: Rewards are minted as new PYRA tokens and distributed at the end of each epoch, aligning participants' incentives with the platform's growth and stability.

4. Impact on Tokenomics:

   • Staking (Burning): Reduces the circulating supply, potentially increasing the token's value.

   • Minting Rewards: Adds to the total supply, maintaining economic balance.