Fixed Rate Tokens Pricing

The value of Fixed Rate Tokens is computed using continuous compounding. The amount of Fixed Rate Tokens that can be obtained in exchange for Underlying Tokens (ex: USDC) is defined as follow:

$$\text{exchangeRate} = e^{\text{tickInterestRate} \cdot \text{timeToMaturity}}$$

Alternatively the Present Value (PV) of Fixed Rate Tokens in Loan Tokens (ex: USDC) is defined as follow:

$$\text{fixedRateTokenPV} = \frac{1}{e^{\text{tickInterestRate} \cdot \text{timeToMaturity}}}$$

Where timeToMaturity corresponds to:

$$\text{timeToMaturity} = \frac{\text{marketMaturityUnixTimestamp} - \text{currentUnixTimestamp}}{31536000}$$

Where 31536000 corresponds to the number of seconds in 365 days.

Examples

// Lend 1,000 USDC at 10% for 1 year.

loanTokens (USDC) = 1_000
timeToMaturity = 1.0
interestRate = 10%
fixedRateTokens = 1_000 * e^(0.10 * 1.0) = 1_105.17

In the example above the user lending 1,000 USDC for 1 year at 10% will receive 1,105.17 Fixed Rate Tokens and will effectively have earned 105.17 USDC at maturity or 10% APY.

// Borrow 1,000 USDC at 5% for 3 months.

loanTokens (USDC) = 1_000
timeToMaturity = 0.25
interestRate = 5%
fixedRateTokens = 1_000 * e^(0.05 * 0.25) = 1_012.58

In the example above the user borrowing 1,000 USDC for 3 months year at 5% will pay 12.58 in USDC interest at maturity or 5% APY.

Held to maturity

If one acquires a Fixed Rate Token with a time to maturity of 1 year at an exchange rate of 0.905 and holds it to maturity the position's net gain is 10.5% (1/0.905-1) or 10% compounded continuously. As long as the fixed rate market doesn't accrue bad debt Fixed Rate Tokens will be worth one unit of Loan Tokens at maturity. Users holding their Fixed Rate Tokens to maturity will therefore received the implied interest rate they locked in.

Mark to market

If a user acquires a Fixed Rate Token with a time to maturity of 1 year at an exchange rate of 0.905 but decides to exit his position early 6 months before maturity he might experience mark to market gains or losses.

Interest rate decrease

If the interest rate decreases from 10% with 1 year left to maturity (0.905 exchange rate) to 5% with 6 months left to maturity (0.975 exchange rate) the user's net gain is 7.7% over a 6 month period which translates into an effective 15% continuously compound rate.

In this example, we can see that although the user lent at 10% fixed for a one year period, by exiting early at a lower interest rate (higher PV) he effectively accrued interest at a rate of 15% annualized. This is because as interest rates decrease the PV of the Fixed Rate Token increases and effectively converges to 1 faster.

Interest rate increase

If the interest rate increases from 10% with 1 year left to maturity (0.905 exchange rate) to 15% with 6 months left to maturity (0.928 exchange rate) the user's net gain is 2.5% over a 6 month period which translates into an effective 5% continuously compound rate.

In this example, we can see that although the user lent at 10% fixed for a one year period, by exiting early at a higher interest rate (lower PV) he effectively accrued interest at a rate of 5% annualized. This is because as interest rates increase the PV of the Fixed Rate Token decreases and effectively takes longer to converge to 1.