Following AIP-26 it was noticed that the interest accrued for the AMPL market above 75% utilization is higher than expected. After investigation the reason for the discrepancy was identified to be the following:

In AAVEs MathUtils there is an over-approximation in the application of the interest rate for computing compounded interest.

### In calculateCompoundedInterest method

Periodic rate (per second) is computed as:

uint256 ratePerSecond = rate / SECONDS_PER_YEAR;

in comparison to the exact computation.

derived from:

.

The deviation in the applied interest rate causes a small over-approximation for APYs <10%, but grows exponentially as can be seen below.

Expected APY | Effective APY | Expected daily rate | Effective daily rate |
---|---|---|---|

4.00% | 4.08% | 0.01% | 0.01% |

5.00% | 5.13% | 0.01% | 0.01% |

10.00% | 10.52% | 0.03% | 0.03% |

50.00% | 64.87% | 0.11% | 0.14% |

100.00% | 171.83% | 0.19% | 0.27% |

200.00% | 638.91% | 0.31% | 0.55% |

1000.00% | 2202543.09% | 0.68% | 2.78% |

10000.00% | 2.69E+43% | 1.30% | 31.52% |

This difference is especially noticeable in the AMPL market, where the configured interest rate can go up to 10,002% at max utilization.

The AAVE Genesis team has been made aware and they will publish more guidance on the discrepancy.

### ARC Rationale

As mentioned in AIP-26, a nonlinear interest curve is more suited for AMPLâ€™s market and potentially other assets on AAVEâ€™s platform. It turns out that this over-approximation produces just such a curve.

The over-approximation mentioned above results in an exponentially growing curve which allows for defining a more suitable interest curve for AMPL. We propose the following parameters for AAVEâ€™s default interest rate strategy, which produce the curve below:

- Optimal utilization = 80%
- Base rate = 1%
- Slope1 = 2%
- Slope2 = 750%

#### APY table above 80% Utilization:

Utilization | APY |
---|---|

80.00% | 3.05% |

81.00% | 49.93% |

82.00% | 118.15% |

83.00% | 217.40% |

84.00% | 361.82% |

85.00% | 571.94% |

86.00% | 877.67% |

87.00% | 1322.50% |

88.00% | 1969.72% |

89.00% | 2911.43% |

90.00% | 4281.60% |

91.00% | 6275.19% |

92.00% | 9175.85% |

93.00% | 13396.29% |

94.00% | 19536.98% |

95.00% | 28471.63% |

96.00% | 41471.48% |

97.00% | 60386.14% |

98.00% | 87906.81% |

99.00% | 127949.14% |

100.00% | 186210.38% |

#### The part of the curve under 80% Utilization:

Rationale:

- Accounting for the over-approximation of the existing Slope2=10,000% interest curve requires reducing the slope2 parameter.
- Taking advantage of the exponential curve allows for:

a. Setting a higher optimal utilization rate.

b. Setting a higher maximum APY.

without creating a too steep of an interest rate increase right above the optimal utilization rate.