> For the complete documentation index, see [llms.txt](https://limitless.gitbook.io/rammlend/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://limitless.gitbook.io/rammlend/products/universal-lending-pool/longzcb-returns.md). # 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](broken://pages/hInDNqXmr1peiSYq7j6i)). $$ \prod g(u\_i) = \prod f(u\_i) + L(\prod f(u\_i) - \prod h(u\_i)) $$ ### 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](broken://pages/oO52nHviXhhFVcDmEBm1). 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](/rammlend/introduction/underwriting-system.md)