# Interest Rate Response

## #

Response CalculationsFor the introduction of the uDEFI a different interest rate response function was chosen as the impression was that the response function of uUSD is both too small in size with respect to the deviations from the target price and too infrequent with the updates. Therefore, the uDEFI will use a different response function for now. After gathering some experience, the next steps will be decided with the community.

# | Asset | Collateral | Update Frequency | Response Function | Deviation for Maximum Response | Maximum Response | Maximum Response per Annum | Maximum Response per Week per Annum |
---|---|---|---|---|---|---|---|---|

1 | uBTC | uUSD | once per 12 hours | linear | 0.10 | +/-1.27E-10 | +0.40% and -0.40% | +5.6% and -5.6% |

2 | uDEFI | uUSD | once per 12 hours | linear | 0.10 | +/-1.27E-10 | +0.40% and -0.40% | +5.6% and -5.6% |

3 | uUSD | tez (XTZ) | once per week | exponential | 0.25 | +/-1.83E-9 | +5.9% and -5.6% | +5.9% and -5.6% |

4 | uUSD | SIRS | once per week | exponential | 0.25 | +/-1.83E-9 | +5.9% and -5.6% | +5.9% and -5.6% |

5 | uXAU | uUSD | once per 12 hours | linear | 0.10 | +/-1.27E-10 | +0.40% and -0.40% | +5.6% and -5.6% |

### #

Response Calculation ExponentialThe uUSD smart contracts calculate the interest rate response in a non-linear fashion. Larger deviations from the target price have a disproportionately larger interest rate response. The response is an exponential function of the price difference up to a defined certain maximum level. This interest rate response is calculated roughly once a week. The respective time span is the interest rate period.

As the interest rates are calculated by second, the actual numbers are very small. The interest rate response currently runs from +1.83E-9 to -1.83E-9, which is roughly equivalent to +5.9% and -5.6% per annum respectively.

### #

Response Calculation LinearThe newer tracker tokens (all except uUSD) calculate the interest rate response in a linear fashion. The response is an linear function of the price difference up to a defined certain maximum level. This interest rate response is calculated roughly once every 12 hours. The respective time span is the interest rate period.

As the interest rates are calculated by second, the actual numbers are very small. The interest rate response currently runs from +1.27E-10 to -1.27E-10, which is roughly equivalent to +0.4% and -0.4% per annum respectively. Per week the maximum of the 14 responses is +5.6% and -5.6%.

## #

Asset Interest RateOn a weekly basis the interest rate response is added to the previously applied per second interest rate. This result is then capped at 8.19E-9 and floored at 1.28E-10. In per annum terms these values are roughly equivalent to +29.4% and +0.4% respectively.

The benefit of this two-step approach is that a resulting equilibrium interest rate can be found by the algorithm without the need to know in advance what it is or the equilibrium needing to be static.

## #

Liability Interest RateThe per second liability interest rate is 3.16E-10 (roughly 1% p.a.) higher than the asset interest rate. On a weekly basis the interest rate response is added to the previously applied per second interest rate. The liability interest rate is capped at 8.512E-9 and floored at 4.44E-10. In per annum terms these values are roughly equivalent to +30.7% and +1.4% respectively.

## #

Translation Interest RatesUsing compounding.

per annum (52 weeks) | per second |
---|---|

25.00% | 7.10E-09 |

24.00% | 6.84E-09 |

23.00% | 6.58E-09 |

22.00% | 6.32E-09 |

21.00% | 6.06E-09 |

20.00% | 5.80E-09 |

19.00% | 5.53E-09 |

18.00% | 5.26E-09 |

17.00% | 4.99E-09 |

16.00% | 4.72E-09 |

15.00% | 4.44E-09 |

14.00% | 4.17E-09 |

13.00% | 3.89E-09 |

12.00% | 3.60E-09 |

11.00% | 3.32E-09 |

10.00% | 3.03E-09 |

9.00% | 2.74E-09 |

8.00% | 2.45E-09 |

7.00% | 2.15E-09 |

6.00% | 1.85E-09 |

5.00% | 1.55E-09 |

4.00% | 1.25E-09 |

3.00% | 9.40E-10 |

2.00% | 6.30E-10 |

1.00% | 3.16E-10 |