> For the complete documentation index, see [llms.txt](https://limitless.gitbook.io/ramm/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/ramm/instrument-examples-and-usage/conditional-lending-pool-clp/longzcb-returns.md).

# longZCB Returns

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. 

{% hint style="info" %}
Note that the returns curve below assumes that the pool stay solvent and no defaults occur. 
{% endhint %}

<figure><img src="https://3340105099-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FFXDxq10tgILHm4JAzNTN%2Fuploads%2FhsohupLpGddgVwj8XUlM%2Ff.png?alt=media&amp;token=6fd62355-21f1-4b86-8dff-4ac0b375f38e" alt=""><figcaption><p>Borrow APR distributed to junior(g) and senior tranche(h) as a function of utilization rate. f(u) is the actual APR paid by borrowers </p></figcaption></figure>

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](/ramm/introduction/whitepaper.md)).&#x20;

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