NAV Formula

The Net Asset Value (NAV) is the core economic invariant of every ERC-20E smart contract. It defines the per-unit collateral backing of the token and establishes a cryptographically enforced, monotonically non-decreasing value floor.

All NAV dynamics are deterministic, fully on-chain, and independent of external price feeds.

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Initial Ratio and Contract Initialization

At deployment, each ERC-20E contract defines an immutable initial issuance ratio r_0, establishing the foundational relationship between token units and collateral units.

$$r_0 = \frac{Collateral\ Units}{Token\ Units}$$

Example:

If the initial ratio is defined as:

$$1 : 0.001$$

then:

• 1 token represents 0.001 units of collateral

• Initial NAV is:

$$NAV_0 = r_0 = 0.001$$

This parameter:

• Defines the starting NAV

• Establishes denomination granularity

• Determines first-mint scaling

• Is immutable post-deployment

After initialization, NAV evolves exclusively according to reserve mechanics.

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NAV Definition

At any time t:

$$NAV_t = \frac{TVL_t}{Supply_t}$$

Where:

• TVL_t = Total collateral held in the contract at time t

• Supply_t = Total outstanding token supply at time t

NAV is:

• Fully on-chain

• Oracle-free

• Deterministic

• Publicly verifiable

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Fee Structure and Accounting Treatment

Let:

• f_p = protocol fee

• f_i = issuer fee

Protocol Fee

• Retained within the reserve

• Increases (TVL)

• Does not increase supply

Issuer Fee

• Transferred externally

• Never retained in reserve

• Timing differs by operation:

  • Mint: deducted before collateral check-in

  • Redemption: deducted after collateral check-out

This sequencing ensures issuer fees do not influence NAV formula.

Minting and NAV Impact

Let a user deposit collateral C.

Step 1: Issuer Fee (Pre Check-In)

$$C_{after\ issuer} = C (1 - f_i)$$

Issuer fee is transferred externally and never enters (TVL).

Step 2: Protocol Fee (Retained)

$$Protocol\ Fee = C_{after\ issuer} \cdot f_p$$

This amount increases (TVL).

Step 3: Net Minting Principal

$$C_{net} = C_{after\ issuer} (1 - f_p)$$

Tokens minted:

$$Tokens_{minted} = \frac{C_{net}}{NAV_{pre}}$$

Post-Mint NAV

$$NAV_{post} = \frac{TVL_{pre} + C_{net} + Protocol\ Fee} {Supply_{pre} + Tokens_{minted}}$$

Since:

$$Protocol\ Fee > 0$$

and supply increases only proportionally to C_{net},

$$NAV_{post} > NAV_{pre}$$

Thus, every mint strictly increases NAV.

Redemption and NAV Impact

Let T tokens be redeemed.

Step 1: NAV-Based Checkout

$$Gross\ Collateral = T \cdot NAV_{pre}$$

Step 2: Protocol Fee Retention

$$Protocol\ Fee = Gross\ Collateral \cdot f_p$$

This portion remains in the reserve.

Step 3: Issuer Fee (Post Check-Out)

$$Issuer\ Fee = Gross\ Collateral \cdot f_i$$

Issuer fee is deducted from user proceeds after NAV calculation.

Step 4: Net Collateral Returned

$$Collateral_{returned} = Gross\ Collateral (1 - f_p - f_i)$$

Post-Redemption NAV

$$NAV_{post} = \frac{TVL_{pre} - Gross\ Collateral + Protocol\ Fee} {Supply_{pre} - T}$$

Because:

• Redemption removes collateral proportionally to NAV

• Protocol fees remain in reserve

• Supply decreases

It follows that:

$$NAV_{post} > NAV_{pre}$$

Thus, every redemption strictly increases NAV.

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Monotonicity Theorem

For every protocol action (mint or redemption):

$$NAV_{t+1} > NAV_t$$

Proof intuition:

1. Collateral withdrawals are strictly proportional to NAV

2. Protocol fees are retained

3. Issuer fees never reduce TVL used in NAV

4. No external price dependency exists

Therefore, NAV is a stepwise strictly increasing function under active protocol usage.

If no activity occurs:

$$NAV_{t+1} = NAV_t$$

NAV can never decrease.