longZCB Returns

The price of longZCB is going to be determined by the interest accrued by the lending pool compared to the promisedReturn parameter where promisedReturn is the returns allocated to the senior tranche(vault) and is a function of the pool's utilization rate.

The more the pool accrues compared to the promisedReturn without incurring bad debt, the higher the yield accrued by longZCB . However, if the pool accrues less yield compared to the promisedReturn(which occurs with unliquidated defaults), longZCB 's APR will be less than the promisedReturn. More information is below.

The returns of longZCB is a monotonically increasing function of the utilization rate of the lending pool. The curve below shows this, where f(u) illustrates the relationship between the utilization rate and the interest rate paid by the borrowers. h(u) is the promisedReturn returned to the senior tranche portion of the supply position and is allocated to the vault holders. g(u) illustrates the pro rata share of the remaining assets after all senior tranche portion has been paid, which are the returns longZCB holders will be entitled to.

The longZCB's return curve as a function of utilization rate g(u) can be represented as the following formula, where the product applies for all timesteps until i, and L is the leverageFactor constant (Derivations in the whitepaper).

$$\prod g(u_i) = \prod f(u_i) + L(\prod f(u_i) - \prod h(u_i))$$

How redemption prices are computed

A user can redeem longZCB anytime there is withdrawable liquidity in its underlying instrument. Redemption prices are calculated as the continuously priced value of the instrument's junior tranche, as illustrated in tradable tranches. At a high level, its pricing equation can be simplified down to the following

$$longZCBPrice =juniorValue = \frac{(totalAssets - (seniorValue * s_s))}{s_j}$$

In words, it is the leftover asset after all senior supply s_s has redeemed for seniorValue. seniorValue is the value accrued to the passive investors of VT, and can be computed via the following equation,

$$seniorValue = inceptionPrice * \prod_t(1+promisedReturn_t)$$

inceptionPrice refers to the starting price of longZCB tokens, and promisedReturn_T refers to the per second compounded rate of promised return for senior holders at time T. Intuitively, the higher the instrument's yield is compared to the promisedReturn_T, the more valuable longZCB becomes.

The P&L realized for the manager would be

$$longZCBAmount * (longZCBPrice_{i+1} - longZCBPrice_{i})$$

Detailed derivations and explanations for the pricings are in the whitepaper