Interest Rate Determination Mechanism: Zero Coupon

• This describes single asset Temporal AMM pool for asset ‘X’

• Let $𝐿_𝑑$ be the number of asset units in the pool which unlock upon completion of the specific period ‘d’ (for duration). Duration is specified in days.
• Let 𝐴𝑑 be the number of assets in the pool which unlock on or after completion of the specific period ‘d’ ($𝐿_𝑑$) a. $A_d = \sum_{d} ^ {d_{max}} L_d$ (from d through to $d_{max}$)
• Let $iX$ be the ‘notional interest units’; these are not real assets, but just book-keeping units created at pool bootstrapping to initialize a desired yield curve.
• Let $iX_d$ be the number of units of $iX$ at d.
• Let $Y_d$ (Yield) be the interest rate at d
• $Y_d = [\frac{A_d + iX_d} {A_d} ^{(\frac {365} {d})}] - 1$
• For any transaction of ‘u’ units at ‘d’, the following occurs

• $L_d \pm u;$ implying $↻_d^z (A_z \pm u).$ where $↻$ represents the function iterates itself from $min$ to $max$ value (here, $d_{min}$ through $d$).
• $↻_{𝑧 = 𝑑_{min}}^{d} (iX \pm i)$; where, ↻ represents the function iterates itself from $min$ to $max$ value (here, $d_{min}$ through $d$).
• $\pm$ are determined by the table below in section ‘Transaction Impact’
• In a borrowing / lending transaction (from pool’s perspective), $'𝑢'$ units are transferred to / from the pool at the time of the transaction, and $[u \times { (1 + Y_d) ^ {(\frac {d} {365})} }]$ are to be repaid by / to the pool at ‘d’