# Borrows

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## Accounting

In Aloe II, the `Lender` is responsible for tracking borrows. This means two things:

* recording the amount each address has borrowed
* accounting for the growth of these borrows as interest accrues

The interest compounds continuously, so borrow growth should be modeled with the exponential function:

{% hint style="info" %}
$$\text{totalBorrows}*{t=T}=\text{totalBorrows}*{t=0} \cdot e^{rT}$$
{% endhint %}

Since there are no borrows at deployment time, and different users request borrows at different times, $$\text{totalBorrows}\_{t=0}$$ is purely hypothetical. When a user takes on some amount of $$\text{newBorrows}$$, we must update the hypothetical value:

{% hint style="info" %}
$$\text{totalBorrows}*{t=T} + \text{newBorrows}=(\text{totalBorrows}*{t=0} + x) \cdot e^{rT}$$

$$\text{totalBorrows}*{t=0} \cdot e^{rT} + \text{newBorrows}=\text{totalBorrows}*{t=0} \cdot e^{rT} + xe^{rT}$$

$$\text{newBorrows}=xe^{rT}$$

$$x=\frac{\text{newBorrows}}{e^{rT}}$$
{% endhint %}

So when a user calls `borrow`, we add $$x$$ to borrows-at-deployment-time, and when a user calls `repay`, we do the opposite: subtract from borrows-at-deployment-time. This addition and subtraction applies *both* to the user's `borrows`, and to the pool's total `borrowBase`.

We'd be done if `r` were constant, but it's not－the interest rate changes with time, as a function of utilization. Instead of an elegant exponential, we get a piecewise function:

{% hint style="info" %}
$$\text{totalBorrows}*{t}= \begin{cases}  \text{totalBorrows}*{t=0} \cdot e^{r\_0 t} & t \leq T\_0 \ \text{totalBorrows}*{t=0} \cdot e^{r\_0 T\_0} e^{r\_1 (t - T\_0)} & T\_0 \lt t \leq T\_1 \ \text{totalBorrows}*{t=0} \cdot e^{r\_0 T\_0} e^{r\_1 (T\_1 - T\_0)} e^{r\_2 (t - T\_1)} & T\_1 \lt t \leq T\_2 \ ... \end{cases}$$
{% endhint %}

The product of this chain of exponentials is the `borrowIndex`. For precision, it starts at $$10^{12}$$ (instead of 1) and it gets updated in the first call to the pool in a given block. Subsequent calls in the same block do nothing because `block.timestamp` is the same. The update looks like:

`borrowIndex *= e ** (currentRate * secondsSinceLastUpdate)`

*In practice we do change-of-base and implement `(1 + yieldPerSecond) ** T`, but the concept is the same.*

As for the coefficient, $$\text{totalBorrows}\_{t=0}$$, we scale that up by $$2^{32}$$ and call it `borrowBase`. Individual users have their own personal `borrowBase`s stored in the `borrows` mapping (we use the expression for $$x$$ from earlier, but scale up by $$2^{32}$$ and divide by `borrowIndex` instead of the plain exponential).

<table><thead><tr><th width="198">Value</th><th>Expression</th></tr></thead><tbody><tr><td>Total Borrows</td><td><code>borrowBase * borrowIndex / (2**32 * 10**12)</code></td></tr><tr><td>User's Borrows*</td><td><code>borrows[user] * borrowIndex / (2**32 * 10**12)</code></td></tr></tbody></table>

*\*In the code, we subtract one from `borrows[user]` because the first unit is used for whitelisting; it's not real debt.*

## Interest Rate

The interest rate is a function of utilization, defined in `RateModel`. The default `RateModel` is a rational function.

{% embed url="<https://www.desmos.com/calculator/tkuyru3pfr>" %}
The <mark style="color:blue;">blue</mark> line shows the borrow APY, and the black line shows the supply APY (assuming a 12.5% reserve factor).
{% endembed %}

The supply rate (earned by lenders) is always less than the borrow rate (paid by borrowers). Given utilization $$U$$ and reserve factor $$f$$, they are related like so:

$$\text{rate}*{supply}=(1 - \frac{1}{f}) \cdot U \cdot \text{rate}*{borrow}$$