Adjust interest rate to account for APY over-approximation

Simple Summary

• Update interest rate on AMPL market to account for over-approximation in compounded interest computation.
• Make use of the exponential curve due to the over-approximation to set higher max APY.

Motivation

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

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.

Specification

As mentioned in AIP-26, a nonlinear - logistic or exponential - interest curve is more suited for AMPL's market and potentially other assets on AAVE's platform.

The over-approximation mentioned above results in an exponential curve which allows for defining a more suitable interest curve for AMPL. We propose the following parameters, which produces 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

1. Accounting for the over-approximation of the existing Slope2=10,000% interest curve requires reducing the slope2 parameter.
2. Taking advantage of the exponential curve allows for: a. Setting a higher optimal utilization rate. b. Setting a higher maximum APY to reduce the potential for utilization being at 100% for extended periods without creating a too steep of an interest rate increase right above the optimal utilization rate.

Implementation

A deployment of the existing implementation of the Interest Strategy will be used, with the following parameters:

optimalUtilizationRate: new BigNumber(0.8).multipliedBy(oneRay).toFixed(),
baseVariableBorrowRate: new BigNumber(0).multipliedBy(oneRay).toFixed(),
variableRateSlope1: new BigNumber(0.02).multipliedBy(oneRay).toFixed(),
variableRateSlope2: new BigNumber(7.5).multipliedBy(oneRay).toFixed(),

https://etherscan.io/address/0x84d1FaD9559b8AC1Fda17d073B8542c8Fb6986dd#readContract

Copyright

Copyright and related rights waived via CC0.