bfUSD Epoch Vesting

This document describes the bfUSD-specific epoch vesting mechanism used by the Horizon (hbfUSD) and Pulsar (pbfUSD) vaults on Ethereum. It does not apply to bfBTC.

The mechanism is designed to:

• Smooth out ratio changes over time.

• Reduce arbitrage opportunities around the exact epoch update timestamp.

Throughout this document:

• StartRatio: the ratio at the start of the current epoch.

• EndRatio: the target ratio for the current epoch.

• CurrentRatio: the effective ratio at a given timestamp within the epoch.

• EpochDuration: fixed at 24 hours (in seconds) for vesting calculations when the ratio increases.

All formulas below apply independently to both Horizon (hbfUSD) and Pulsar (pbfUSD) vaults.

1. Linear ratio vesting when the epoch ratio increases

When an epoch update results in an increased ratio:

1. After updateEpoch is executed:

   • The previous epoch's EndRatio is stored as the current epoch's StartRatio.

   • The new target ratio for this epoch is stored as EndRatio.

2. EpochDuration is set to 24 hours (in seconds).

3. Over the next 24 hours, the ratio is incremented every second from StartRatio toward EndRatio until, at the end of the 24th hour, it reaches EndRatio.

Let:

• TimeElapsed be the number of seconds elapsed since the epoch start, clamped to [0, EpochDuration].

Then:

$$\mathrm{CurrentRatio} = \mathrm{StartRatio} + \left( \mathrm{EndRatio} - \mathrm{StartRatio} \right) \times \frac{\mathrm{TimeElapsed}}{\mathrm{EpochDuration}}$$

2. Immediate ratio change when the epoch ratio decreases

When an epoch update results in a decreased ratio:

• The protocol does not apply linear vesting.

• It does not use EpochDuration for this case.

• Instead, the epoch ratio is updated immediately to EndRatio at the time of the epoch update.

In other words, for a decreasing ratio, CurrentRatio is set directly to EndRatio with no time-based interpolation.

3. Stake and unstake calculations

This section summarizes how user deposits (stake) and withdrawals (unstake) are calculated under the epoch vesting mechanism.

3.1 Stake (bfUSD -> shares)

When a user stakes bfUSD into Horizon or Pulsar, they receive vault shares. The number of shares is calculated using the final ratio of the current epoch, EndRatio:

• Stake:

  $$\mathrm{ReceiveAmount} = \frac{\mathrm{StakeBfusdAmount}}{\mathrm{EndRatio}}$$

Where:

• StakeBfusdAmount is the amount of bfUSD deposited.

• ReceiveAmount is the number of shares minted to the user.

3.2 Unstake (shares -> bfUSD)

When a user unstakes shares, they receive bfUSD. The amount of bfUSD is calculated using the time-dependent CurrentRatio, which follows the interpolation described in section 1 (for increasing epochs) or is equal to EndRatio (for decreasing epochs):

• Unstake:

  $$\mathrm{ReceiveAmount} = \mathrm{UnstakeSharesAmount} \times \mathrm{CurrentRatio}$$

Where:

• UnstakeSharesAmount is the number of shares burned.

• ReceiveAmount is the amount of bfUSD returned to the user.

• CurrentRatio is given by:

  $$\mathrm{CurrentRatio} = \mathrm{StartRatio} + \left( \mathrm{EndRatio} - \mathrm{StartRatio} \right) \times \frac{\mathrm{TimeElapsed}}{\mathrm{EpochDuration}}$$

3.3 User impact

Under this design:

• If a user stakes and then unstakes within the same epoch while the ratio is increasing, they will incur a loss.

• The approximate loss (in bfUSD terms) is:

  $$\text{Loss} \approx \mathrm{StakeBfusdAmount} \times \left( \mathrm{EndRatio} - \mathrm{CurrentRatio} \right)$$

This behavior:

• Removes profitable short-term "in-and-out" arbitrage based on the linear vesting curve.

• Encourages longer-term holding of pbfUSD and hbfUSD rather than rapid churn around epoch boundaries.