# RAI Controller: Absolute vs Percent Error

This post will detail a nuance in how the RAI system controller calculates the redemption rate and will propose a slight change. The system currently uses the *absolute error* of the market price to compute rates, but using the *percent error* will produce consistent rates under RAI/USD price changes.

For a refresher on how the RAI controller works, go [here](https://medium.com/reflexer-labs/stability-without-pegs-8c6a1cbc7fbd).

## Rate Calcuation

The mainnet RAI controller is currently only a P-controller, so the redemption rate is calculated as:

$redemption\_rate = Kp * (redemption\_price - market\_price)$

where:
$Kp=7.5e^{-8}$
and
$error = redemption\_price - market\_price$

Since the error term is the absolute error between redemption and market prices, the resulting rate is dependent on the the RAI/USD price.

In other words, a 1% market fluctuation will produce different rates in the cases of the RAI/USD redemption being $2, $3, $4, etc.

## Example

To demonstrate the issue with using absolute error, consider the two cases below, each with a different starting redemption price.

Each case shows the redemption and market prices(top), the absolute error(middle), and the resulting redemption rate(bottom).

You can see while both market prices experience fluctations of 1%, the right case produces much higher rates because of the larger redemption price.


## Implications

If RAI/USD was to drift considerable from it's current starting point, the system will produce rates of a much different magnitude. Here are some possible values using the current system $Kp$ of $7.5e^{-8}$.

| Redemption Price | Market deviation | Annual Redemption Rate|
|---------|----------|---------|
|$1|1%| 2.4%|
|$2|1%| 4.8%|
|$3|1%| 7.4%|
|$4|1%| 10.0%|
|$5|1%| 12.6%|
|$6|1%| 15.2%|

## Solution

Instead of absolute error, the RAI controller can use percent error to calculate the market error.

$pct\_error = \displaystyle \frac{redemption\_price - market\_price}{redemption\_price}$

This will produce the same rates for % market deviations, irrespective of redemption price.

## Rates Produced From Pct Error

Below are the same two cases aboves, but with the rate calculated using the percent error.

### But first, one caveat

Using the percent error changes the units of the error, so $Kp$ has to be adjusted if we want to produce similar rates.

If we want to match the above case rate behavior, we simply need to multiply the original $Kp$ by $3$.

$rate_{pct} = rate_{absolute}$

$Kp_{new} * pct\_error = Kp * absolute\_error$

$Kp_{new} * 0.01 = Kp * 0.03$

$Kp_{new} = \displaystyle \frac{Kp * 0.03}{0.01}$

$Kp_{new} = 3 * Kp = 3 * 7.5e^{-8} = 2.25e^{-7}$

And now the two previous cases, but with percent error. The rates are equal.


When using percent error, the rates will be equal for equal % market deviations, no matter what the redemption price is.

| Redemption Price | Market deviation | Annual Redemption Rate|
|---------|----------|---------|
|$1|1%| 7.4%|
|$2|1%| 7.4%|
|$3|1%| 7.4%|
|$4|1%| 7.4%|
|$5|1%| 7.4%|
|$6|1%| 7.4%|